The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime

Abstract

We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-\frac {1}{3}}$. In the limit $N\rightarrow \infty $, we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.

1 Introduction

In the 1940s, experiments on the cohesive energy and specific heat of alkali atoms ¹ showed a large discrepancy with theoretical calculations based solely on the Hartree–Fock approximation [3], further complicated by the fact that second-order perturbation theory failed because it yielded infinities. Motivated by this unfortunate situation, Bohm and Pines in four seminal papers [11, 12, 13, 32] introduced the random phase approximation (RPA) as a useful tool for studying the properties of a high-density electron gas moving in a background of uniform positive charge, called jellium. In the Bohm–Pines RPA approach, the electron gas could be decoupled into collective plasmon excitations and quasi-electrons that interacted via a screened Coulomb interaction. The latter fact justified the independent particle approach commonly used for many-body fermion systems. Their work was also in good agreement with experimental data, the culmination of which was the experimental detection of plasmons [42, 17].

The microscopic derivation of the RPA has led to notable work by theoretical physicists since the 1950s. In 1957, Gell-Mann and Brueckner [20] derived the correlation energy of the electron gas in the high density limit by using a formal summation of a particular class of Feynman diagrams. Although each diagram is divergent in itself, it turned out that the sum is finite. This diagrammatic picture further suggested that the main contribution to the ground-state energy came from the interaction of pairs of fermions, one from inside and one from outside the Fermi ball. Shortly thereafter, Sawada [36] and Sawada–Brueckner–Fukuda–Brout [37] interpreted these pairs as bosons and obtained the correlation energy by diagonalizing an effective Hamiltonian which is quadratic with respect to the bosonic particle pairs. Since then, the random phase approximation has become a cornerstone in the physics of condensed matter and nuclear physics [34], also playing a significant role in bosonic field theory [26], in the quark-gluon plasma [41] and especially in computational chemistry and materials science. Although originally proposed for an electron gas, it is applicable to a wide variety of fermionic systems.

The complete derivation of the RPA from first principles, namely from the microscopic Schrödinger equation, has, however, long been a major open problem in mathematical physics. Recently, some rigorous results on the correlation energy have been derived in the mean-field regime for small interaction potentials by Hainzl–Porta–Rexze [24] (perturbative results) and by Benedikter–Nam–Porta–Schlein–Seiringer [4, 5, 6] (non-perturbative results).

The aim of the present paper is to justify the RPA for a large class of interaction potentials in the mean-field regime, addressing not only the ground state energy but also the excitation spectrum. As we will explain below, the correlation structure of Fermi gases can indeed be described correctly by treating appropriate pairs of fermions as bosons. The corresponding bosonic Hamiltonian can be handled by Bogolubov’s diagonalization method, thus putting the description in the physics literature [20, 36, 37] on a firm mathematical footing. Although this general point of view has been employed in [24, 5, 6], we will provide a new bosonization approach to fermionic systems which enables us to not only extend the study on the ground state energy initiated in [24, 5, 6] but also obtain all bosonic elementary excitations predicted in the physics literature, thus justifying the RPA in the mean-field regime. In the long run, we expect that the tools developed in our work will pave the way towards the Coulomb gas in the thermodynamic limit.

1.1 Model

We consider a system of N (spinless) fermions on the torus $\mathbb {T}^{3}=\left[0,2\pi \right]^{3}$ (with periodic boundary conditions), interacting via a bounded potential $V:\mathbb {T}^{3}\rightarrow \mathbb {R}$ . The system is described by the Hamiltonian

(1.1) $$ \begin{align} H_{N}=H_{\text{kin}}+k_{F}^{-1}H_{\text{int}} = \sum_{i=1}^{N} (-\Delta_{i}) + k_F^{-1}\sum_{1\leq i<j\leq N}V\left(x_{i}-x_{j}\right), \end{align} $$

which acts on the fermionic space

(1.2) $$ \begin{align} \mathcal{H}_{N}=\bigwedge^{N}\mathfrak{h}, \quad \mathfrak{h}=L^{2} (\mathbb{T}^{3} ). \end{align} $$

Here, the coupling constant $k_F^{-1}>0$ corresponds to the interaction strength. We will focus on the mean-field regime $k_F^{-1}\sim N^{-\frac 1 3}$ , where the kinetic and interaction energies are comparable. More precisely, we assume that

(1.3) $$ \begin{align} N=\left|B_{F}\right| = \frac{4\pi}{3} k_F^3 (1+ o(1)_{k_F\to \infty}), \quad B_{F}=\overline{B}(0,k_{F})\cap\mathbb{Z}^{3}, \end{align} $$

namely, the Fermi ball $B_F$ is completely filled by N integer points. In this case, the kinetic operator $H_{\text {kin}}$ has a unique, non-degenerate ground state which is the Fermi state

(1.4) $$ \begin{align} \psi_{\mathrm{FS}}=\bigwedge_{p\in B_{F}}u_{p}, \quad u_{p} (x )=(2\pi)^{-\frac{3}{2}}e^{ip\cdot x}. \end{align} $$

More generally, the eigenstates of $H_{\text {kin}}$ can be written explicitly in terms of the plane waves $\left(u_{p}\right)_{p\in \mathbb {Z}^{3}}$ . However, the spectrum of the interacting operator $H_N$ is highly nontrivial, and its computation often requires suitable approximations.

We assume that V is of positive type, namely, its Fourier transform satisfies $\hat V \ge 0$ with

(1.5) $$ \begin{align} V (x )=\frac{1}{(2\pi)^{3}}\sum_{k\in\mathbb{Z}^{3}}\hat{V}_{k}e^{ik\cdot x}\quad\text{with}\quad\hat{V}_{k}=\int_{\mathbb{T}^{3}}V (x )e^{-ik\cdot x}\,dx. \end{align} $$

Under our assumption, $H_{N}$ is a self-adjoint operator on $\mathcal {H}_{N}$ with domain $D\left(H_{N}\right)=D\left(H_{\text {kin}}\right)=\bigwedge ^{N}H^{2}\left(\mathbb {T}^{3}\right).$ Moreover, $H_N$ is bounded from below and has compact resolvent. We are interested in the asymptotic behavior of the low-lying spectrum of $H_N$ when $N\to \infty $ and $k_F\to \infty $ .

One of the most famous approximations for fermions is the Hartree–Fock theory, where one restricts the states under consideration to the set of all Slater determinants $g_{1}\wedge g_{2}\cdots \wedge g_{N}$ with $\left\{ g_{i}\right\} _{i=1}^{N}$ orthonormal in $L^{2}\left(\mathbb {T}^{3}\right)$ . The precision of the Hartree–Fock energy is an interesting subject, which has been studied for Coulomb systems by Bach [1] and Graf–Solovej [22]. In general, the Hartree–Fock minimizer could be different from the Fermi state $\psi _{\mathrm {FS}}$ ; see [21] for an estimate for Coulomb systems. However, in the mean-field model that we are considering here, the Hartree–Fock minimizer coincides with $\psi _{\mathrm {FS}}$ ; see [6, Theorem A.1] for a precise statement. Thus, to obtain the correction to the ansatz of plane waves, we have to understand the correlation structure of the system. ²

To go beyond the ansatz of plane waves, the first step is the extraction of the energy of the Fermi state. For computational purposes, it is convenient to use the second quantization language. For every $p\in \mathbb {Z}^3$ , we denote by $c_{p}^{\ast }=c^*(u_p)$ , $c_{p}=c(u_p)$ the fermionic creation and annihilation operators associated to the plane-wave state $u_{p}$ . These operators act on the fermionic Fock space

(1.6) $$ \begin{align} \mathcal{F}^{-}\left(\mathfrak{h}\right)=\bigoplus_{N=0}^{\infty}\bigwedge^{N}\mathfrak{h} \end{align} $$

and obey the canonical anticommutation relations (CAR)

(1.7) $$ \begin{align} \left\{ c_{p},c_{q}\right\} =\left\{ c_{p}^{\ast},c_{q}^{\ast}\right\} =0,\quad\left\{ c_{p},c_{q}^{\ast}\right\} =\delta_{p,q},\quad p,q\in\mathbb{Z}^{3}, \end{align} $$

where $\left\{ A,B\right\} =AB+BA$ . The Hamiltonian operator $H_{N}$ in (1.1) can be expressed as

(1.8) $$ \begin{align} H_{N}=H_{\text{kin}}+k_{F}^{-1}H_{\text{int}} = \sum_{p\in\mathbb{Z}^{3}}|p|^{2}c_{p}^{\ast}c_{p}+\frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}^{3}}\sum_{p,q\in\mathbb{Z}^{3}}\hat{V}_{k}c_{p+k}^{\ast}c_{q-k}^{\ast}c_{q}c_{p}. \end{align} $$

Thanks to the CAR (1.7), it is straightforward to see that the Fermi state obeys, for all $p\in \mathbb {Z}^{3}$ ,

(1.9) $$ \begin{align} c_{p}^{\ast}c_{p}\psi_{\mathrm{FS}}=1_{B_{F}}(p)\psi_{\mathrm{FS}}=\begin{cases} \psi_{\mathrm{FS}} & p\in B_{F}\\ 0 & p\in B_{F}^{c}, \end{cases} \end{align} $$

where $1_{B_{F}}(\cdot )$ denotes the indicator function of the Fermi ball $B_{F}$ . Thus, the kinetic energy of the Fermi state is

(1.10) $$ \begin{align} \left\langle \psi_{\mathrm{FS}},H_{\text{kin}}\psi_{\mathrm{FS}}\right\rangle =\sum_{p\in\mathbb{Z}^{3}}|p|^{2}\left\langle \psi_{\mathrm{FS}},c_{p}^{\ast}c_{p}\psi_{\mathrm{ FS}}\right\rangle =\sum_{p\in\mathbb{Z}^{3}}1_{B_{F}}(p)|p|^{2}\left\Vert \psi_{\mathrm{ FS}}\right\Vert ^{2}=\sum_{p\in B_{F}}|p|^{2}. \end{align} $$

Hence, we can define the localized kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }:D\left(H_{\text {kin}}\right)\subset \mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ by

(1.11) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime}=H_{\text{kin}} - \left\langle \psi_{\mathrm{FS}},H_{\text{kin}}\psi_{\mathrm{FS}}\right\rangle = \sum_{p\in B_{F}^{c}}|p|^{2}c_{p}^{\ast}c_{p}-\sum_{p\in B_{F}}|p|^{2}c_{p}c_{p}^{\ast}. \end{align} $$

We refer to this operator as being ‘localized’ since extracting $\langle \psi _{\mathrm {FS}} , H_{\mathrm {kin}} \psi _{\mathrm {FS}}\rangle $ in this manner can be seen as changing the point of reference from the vacuum state $\Omega $ to the Fermi state $\psi _{\mathrm {FS}}$ , so $H_{\mathrm {kin}}^\prime $ can be seen as a kind of expansion of $ H_{\mathrm {kin}}$ around $\psi _{\mathrm {FS}}$ .

Note that it is clear from the first identity in (1.11) that $H_{\operatorname {\mathrm {kin}}}^{\prime }$ is nonnegative since $\psi _{\mathrm {FS}}$ is the ground state of $H_{\text {kin}}$ . However, the positivity of $H_{\operatorname {\mathrm {kin}}}^{\prime }$ is unclear from the second identity in (1.11) since the difference of two operators which are nonnegative may not have a sign. The resolution of this apparent paradox lies in the underlying Hilbert space: in the N-body space $\mathcal {H}_{N}$ , we always have

(1.12) $$ \begin{align} N = \sum_{p\in\mathbb{Z}^{3}}c_{p}^{\ast}c_{p}=\sum_{p\in B_{F}} (1- c_{p} c_{p}^{\ast}) +\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}= |B_F|- \sum_{p\in B_{F}}c_{p}c_{p}^{\ast}+\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}. \end{align} $$

Therefore, the assumption $|B_F|=N$ implies the particle-hole symmetry

(1.13) $$ \begin{align} \mathcal{N}_{E}=\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}=\sum_{p\in B_{F}}c_{p}c_{p}^{\ast}\quad\text{on }\mathcal{H}_{N}, \end{align} $$

namely, the excitation number operator (which counts the number of particles outside the Fermi state) coincides with the hole number operator (which counts the number of holes inside the Fermi state). Consequently, the kinetic operator in (1.11) can be rewritten as

(1.14) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime} =\sum_{p\in B_{F}^{c}}\vert|p|^{2}-\zeta \vert\,c_{p}^{\ast}c_{p}+\sum_{p\in B_{F}}\vert|p|^{2}-\zeta\vert\,c_{p}c_{p}^{\ast} \end{align} $$

for any $\zeta \in [\sup _{p\in B_{F}}|p|^{2},\inf _{p\in B_{F}^{c}}|p|^{2}]$ , which is clearly nonnegative.

For the interaction operator, it is convenient to use the factorized form

(1.15) $$ \begin{align} H_{\text{int}} &=\frac{1}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}^{3}}\sum_{p,q\in\mathbb{Z}^{3}}\hat{V}_{k}c_{p+k}^{\ast}c_{q-k}^{\ast}c_{q}c_{p} \nonumber \\ &=\frac{1}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}^{3}}\hat{V}_{k}(\text{d}\Gamma(e^{-ik\cdot x})^{\ast}\text{d}\Gamma(e^{-ik\cdot x})-N), \end{align} $$

where

(1.16) $$ \begin{align} \text{d}\Gamma(e^{-ik\cdot x})=\sum_{p,q\in\mathbb{Z}^{3}}\left\langle u_{p},e^{-ik\cdot x}u_{q}\right\rangle c_{p}^{\ast}c_{q}=\sum_{p,q\in\mathbb{Z}^{3}}\delta_{p,q-k}c_{p}^{\ast}c_{q}=\sum_{p\in\mathbb{Z}^{3}}c_{p}^{\ast}c_{p+k}. \end{align} $$

Note that for any $k\in \mathbb {Z}_{\ast }^{3}=\mathbb {Z}^{3}\backslash \left\{ 0\right\}$ , we have

(1.17) $$ \begin{align} \text{d}\Gamma(e^{-ik\cdot x})\psi_{\mathrm{FS}}=\sum_{p\in\mathbb{Z}^{3}}c_{p}^{\ast}c_{p+k}\psi_{\mathrm{FS}}=\sum_{p\in L_{-k}}c_{p}^{\ast}c_{p+k}\psi_{\mathrm{FS}} \end{align} $$

since the summand $c_{p}^{\ast }c_{p+k}\psi _{\mathrm {FS}}$ in (1.17) does not vanish if and only if $p\in L_{-k}$ , where the lune

(1.18) $$ \begin{align} L_k = B_{F}^{c}\cap\left(B_{F}+k\right)= \left\{ p\in\mathbb{Z}^{3}\mid|p-k|\leq k_{F}<|p|\right\} \end{align} $$

will play an important role in our analysis. In particular, using (1.9) and the CAR again, we find that for all $k\in \mathbb {Z}_{\ast }^{3}$ ,

(1.19) $$ \begin{align} \left\Vert \text{d}\Gamma(e^{-ik\cdot x})\psi_{\mathrm{FS}}\right\Vert ^{2}=\sum_{p\in L_{-k}}\left\Vert c_{p}^{\ast}c_{p+k}\psi_{\mathrm{FS}}\right\Vert ^{2}=\sum_{p\in L_{-k}}1=\left|L_{-k}\right|=\left|L_{k}\right|. \end{align} $$

Thus, the interaction energy of the Fermi state is given by

(1.20) $$ \begin{align} \left\langle \psi_{\mathrm{FS}},H_{\text{int}}\psi_{\mathrm{FS}}\right\rangle =\frac{N\left(N-1\right)}{2(2\pi)^{3}}\hat{V}_{0}+\frac{1}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(\left|L_{k}\right|-N\right), \end{align} $$

where we see the direct and exchange energies (involving $\hat V_0$ and $\{\hat V_k\}_{k\ne 0}$ , respectively). We can define the localized interaction operator

(1.21) $$ \begin{align} H_{\text{int}}^{\prime}=H_{\text{int}}- \left\langle \psi_{\mathrm{FS}},H_{\text{int}}\psi_{\mathrm{ FS}}\right\rangle = \frac{1}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}(\text{d}\Gamma(e^{-ik\cdot x})^{\ast}\text{d}\Gamma(e^{-ik\cdot x})-\left|L_{k}\right|). \end{align} $$

In summary, with $H_{\operatorname {\mathrm {kin}}}^{\prime }$ and $H_{\operatorname {\mathrm {int}}}^{\prime }$ defined in (1.11) and (1.21), we can write

(1.22) $$ \begin{align} H_{N} =E_{\mathrm{FS}} +H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1}H_{\operatorname{\mathrm{int}}}^{\prime}, \quad E_{\mathrm{FS}} = \left\langle \psi_{\operatorname{\mathrm{FS}}},H_{N}\psi_{\operatorname{\mathrm{FS}}}\right\rangle. \end{align} $$

Note that in the prior works [24, 5, 6], the localization procedure was carried out by employing what is known as the particle-hole transformation, which maps the Fermi state $\psi _{\mathrm {FS}}$ to the vacuum; see, for example, [6, Eq. (1.20)] for an analogue of (1.22). However, in the present paper we do not follow this approach since we prefer to work on the N-body Hilbert space.

1.2 Random phase approximation

In this subsection, we explain the ideas of the bosonization approach to the random phase approximation. On the one hand, in the original approach [11, 12, 13, 32], Bohm and Pines considered fluctuations of density in the momentum representation where the plasma momenta and the effective particle momenta of different wavelengths $k,l$ are coupled by phases $e^{i(k-l) \cdot x_j}$ , summing over the ‘random’ particle positions $x_j$ . The assumption that the phases average toward zero for a large number of particles is originally called the ‘random phase approximation’. On the other hand, after the work of Sawada [36] and Sawada–Brueckner–Fukuda–Brout [37], the term RPA has been widely used in the physics literature in the context of a quasi-bosonic Hamiltonian, where a quasi-boson consists of a particle-hole pair. The quasi-bosonic approach is used not only for Coulomb gases but also in a much broader context, especially in nuclear matter (for a standard textbook, see [18, p. 156] for Coulomb gases and [18, pp. 540-543] for nuclear matter).

In the present paper, we will focus on building a mathematical formulation of the quasi-bosonic approach for general potentials and eventually apply this theory to regular potentials. In the long run, we hope that this general theory will also be helpful for singular potentials, in particular for Coulomb gases where the next-order correction to the bosonization picture matters (in [15], we used the formulation provided in the present paper to find the analogue of the Gell-Mann–Brueckner formula for the mean-field Coulomb gas, which shows how important it is to carry the non-bosonic part in the calculation at least to the leading order).

Now let us explain the bosonization argument in detail. Roughly speaking, the RPA suggests that the fermionic correlation can be described by a Hamiltonian which is quadratic in suitable bosonic creation and annihilation operators. To explain the heuristic bosonization argument, let us decompose further the interaction terms in (1.21) by defining, for every $k\in \mathbb {Z}_{\ast }^{3}$ ,

(1.23) $$ \begin{align} \text{d}\Gamma\big(e^{-ik\cdot x}\big)=\text{d}\Gamma\big(\big(P_{B_{F}}+P_{B_{F}^{c}}\big)e^{-ik\cdot x}\big(P_{B_{F}}+P_{B_{F}^{c}}\big)\big) = \tilde{B}_{k}+\tilde{B}_{-k}^{\ast}+D_{k}, \end{align} $$

where $P_{B_F}$ and $P_{B_F^c}$ are projections in the one-fermion Hilbert space and

(1.24) $$ \begin{align} \tilde{B}_{k}&=\text{d}\Gamma\big(P_{B_{F}}e^{-ik\cdot x}P_{B_{F}^{c}}\big)=\sum_{p,q\in\mathbb{Z}^{3}}\left\langle u_{p}, P_{B_{F}}e^{-ik\cdot x}P_{B_{F}^{c}} u_{q}\right\rangle c_{p}^{\ast}c_{q} = \sum_{p\in L_{k}}c_{p-k}^{\ast}c_{p}, \\ D_k &= \text{d}\Gamma\big(P_{B_{F}}e^{-ik\cdot x}P_{B_{F}}\big) + \text{d}\Gamma\big(P_{B_{F}^c}e^{-ik\cdot x}P_{B_{F}^{c}}\big) = \sum_{p\in B_F \cap (B_F+k)} c_{p-k}^{\ast}c_{p} + \sum_{p\in B_F^c \cap (B_F^c+k) } c_{p-k}^{\ast}c_{p}. \nonumber \end{align} $$

Note that for all $k\in \mathbb {Z}^3_{*}$ , we have $D_k^*=D_{-k}$ and

(1.25) $$ \begin{align} [\tilde{B}_k,\tilde{B}_{-k}]= [\tilde{B}_{-k},D_{k}]=[\tilde{B}_{k}^{\ast},D_{k}]=0, \end{align} $$

which can be seen from the identity $[\mathrm {d} \Gamma (X),\mathrm {d} \Gamma (Y) ] = \mathrm {d} \Gamma ([X,Y])$ and (1.24). Due to the symmetry between k and $-k$ , it is convenient to introduce the set ³

(1.26) $$ \begin{align} \mathbb{Z}_{+}^{3}=\left(\big\{ x_{1}>0\big\} \cup\left\{ x_{1}=0,x_{2}>0\right\} \cup\left\{ x_{1}=x_{2}=0,x_{3}>0\right\} \right)\cap\mathbb{Z}_{\ast}^{3} \end{align} $$

such that

(1.27) $$ \begin{align} \mathbb{Z}_{+}^{3}\cup\big(-\mathbb{Z}_{+}^{3}\big)=\mathbb{Z}_{\ast}^{3}, \quad \mathbb{Z}_{+}^{3}\cap\big(-\mathbb{Z}_{+}^{3}\big)=\emptyset. \end{align} $$

Using this notation and the assumption $\hat V_k=\hat V_{-k}$ , we can rewrite the interaction operator in (1.21) as

(1.28) $$ \begin{align} k_{F}^{-1} H_{\text{int}}^{\prime} & =\frac{k_F^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}} \hat{V}_{k}\left(\left(\tilde{B}_{k}+\tilde{B}_{-k}^{\ast}+D_{k}\right)^{\ast}\left(\tilde{B}_{k}+\tilde{B}_{-k}^{\ast}+D_{k}\right)-\left|L_{k}\right|\right) \nonumber\\ & = \sum_{k\in\mathbb{Z}_{+}^{3}}\left(H_{\text{int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right) +\frac{k_{F}^{-1}}{(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(\tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k}+\frac{1}{2}D_{k}^{\ast}D_{k}\right) , \end{align} $$

where for each $k\in \mathbb {Z}_{+}^{3}$ , we denote

(1.29) $$ \begin{align} H_{\text{int}}^{k} & =\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left(\left(\tilde{B}_{k}+\tilde{B}_{-k}^{\ast}\right)^{\ast}\left(\tilde{B}_{k}+\tilde{B}_{-k}^{\ast}\right)+\left(\tilde{B}_{-k}+\tilde{B}_{k}^{\ast}\right)^{\ast}\left(\tilde{B}_{-k}+\tilde{B}_{k}^{\ast}\right)\right) \nonumber \\ & =\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}} \left( \{ \tilde{B}_{k}^{\ast}, \tilde{B}_{k}\} + \{\tilde{B}_{-k}^{\ast}, \tilde{B}_{-k}\} + 2 \tilde{B}_{k}^{\ast} \tilde{B}_{-k}^{\ast} +2 \tilde{B}_{-k} \tilde{B}_{k} \right). \end{align} $$

Now let us introduce the quasi-bosonicity. From the CAR (1.7), it is straightforward to see that

(1.30) $$ \begin{align} \left[\tilde{B}_{k},\tilde{B}_{l}\right]=\left[\tilde{B}_{k}^{\ast},\tilde{B}_{l}^{\ast}\right]=0,\quad \left[\tilde{B}_{k},\tilde{B}_{l}^{\ast}\right]= \left|L_{k}\right|\delta_{k,l}-\sum_{p\in L_{k}\cap L_{l}}c_{p-l}c_{p-k}^{\ast}-\sum_{p\in L_{k}\cap\left(L_{l}-l+k\right)}c_{p-k+l}^{\ast}c_{p} \end{align} $$

for all $k,l\in \mathbb {Z}^3_{*}$ , where $[A,B]= AB-BA$ . Hence, on states with few excitations (e.g., the expectation value of $\mathcal {N}_E$ is much smaller than $|L_k| \sim \min \{|k| k_F^2,k_F^3\}$ ), the rescaled operators $\tilde {B}_{k}^{\prime }=\left|L_{k}\right|^{-\frac {1}{2}}\tilde {B}_{k}$ obey the commutation relations

(1.31) $$ \begin{align} \left[\tilde{B}_{k}^{\prime},\tilde{B}_{l}^{\prime}\right] =\left[\left(\tilde{B}_{k}^{\prime}\right)^{\ast},\left(\tilde{B}_{l}^{\prime}\right)^{\ast}\right]=0, \quad \left[\tilde{B}_{k}^{\prime},\left(\tilde{B}_{l}^{\prime}\right)^{\ast}\right] \approx \delta_{k,l} \end{align} $$

for all $k,l\in \mathbb {Z}_{\ast }^{3}$ , in direct analogy with the canonical commutation relations (CCR) obeyed by a set of bosonic creation and annihilation operators $a_{k}^{\ast }$ , $a_{k}$ indexed by $\mathbb {Z}_{\ast }^{3}$ ,

(1.32) $$ \begin{align} \left[a_{k},a_{l}\right] =\left[a_{k}^{\ast},a_{l}^{\ast}\right]=0, \quad \left[a_{k},a_{l}^{\ast}\right] =\delta_{k,l}. \end{align} $$

Since the relation $[\tilde {B}_{k}^{\prime }, (\tilde {B}_{l}^{\prime })^{\ast }] \approx \delta _{k,l}$ is only approximate, we call these operators quasi-bosonic.

In view of the quasi-bosonicity of these operators, in the form (1.28) of $H_{\text {int}}^{\prime }$ , we call the first sum on the right-hand side of this equation the bosonizable terms, while the second sum constitutes the non-bosonizable terms which are regarded as error terms. The bosonizable part $H_{\text {int}}^{k}$ can be viewed as a quadratic Hamiltonian in the bosonic setting, which can be diagonalized by Bogolubov transformations. This is the spirit of what we will do, but there is a catch: the kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ cannot be written in terms of $\tilde {B}_{k}$ . The solution is to further decompose the operators $\tilde {B}_{k}$ by defining the excitation operators

(1.33) $$ \begin{align} b_{k,p}=c_{p-k}^{\ast}c_{p},\quad b_{k,p}^{\ast}=c_{p}^{\ast}c_{p-k},\quad k\in\mathbb{Z}_{\ast}^{3},\,p\in L_{k}. \end{align} $$

The name is due to the fact that the action of $b_{k,p}^{\ast }$ is to create a state at momentum $p\in B_{F}^{c}$ and annihilate a state at momentum $p-k\in B_{F}$ .

Since $H_{\text {int}}^{k}$ is quadratic in terms of $\tilde {B}_{k}$ , it is also quadratic in terms of $b_{k,p}^{\ast }$ , namely,

(1.34) $$ \begin{align} H_{\text{int}}^{k} & =\sum_{p,q\in L_{k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\big(b_{k,p}^{\ast}b_{k,q}+b_{k,q}b_{k,p}^{\ast}\big)+\sum_{p,q\in L_{-k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\big(b_{-k,p}^{\ast}b_{-k,q}+b_{-k,q}b_{-k,p}^{\ast}\big)\nonumber \\ &\quad +\sum_{p\in L_{k}}\sum_{q\in L_{-k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\big(b_{k,p}^{\ast}b_{-k,q}^{\ast}+b_{-k,q}b_{k,p}\big)+\sum_{p\in L_{-k}}\sum_{q\in L_{k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\big(b_{-k,p}^{\ast}b_{k,q}^{\ast}+b_{k,q}b_{-k,p}\big). \end{align} $$

The reason that the operators $b_{k,p}$ are preferable to the operators $\tilde {B}_{k}$ is that they satisfy the following commutation relation with the kinetic operator (see (1.74) below)

(1.35) $$ \begin{align} \big[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{k,p}^{\ast}\big]= 2 \lambda_{k,p} b_{k,p}^{\ast}, \quad \lambda_{k,p}=\frac{1}{2}(|p|^{2}-|p-k|^{2}). \end{align} $$

Note that $\lambda _{k,p}\ge \frac {1}{2}$ (first, $\lambda _{k,p}>0$ since $p\in L_k$ ; moreover, $|p|^2- |p-k|^2$ is an integer as $p,k\in \mathbb {Z}^3$ ). This is to be compared with the bosonic setting: if the operators $a_{k}$ obey the CCR (1.32), then

(1.36) $$ \begin{align} \left[\sum_{l}\varepsilon_{l}a_{l}^{\ast}a_{l},a_{k}^{\ast}\right]=\varepsilon_{k}a_{k}^{\ast}. \end{align} $$

Therefore, viewing $b_{k,p}^{\ast }$ as being analogous to a bosonic creation operator, we get

(1.37) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime}\approx \sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}} 2 \lambda_{k,p} b_{k,p}^{\ast}b_{k,p} =\sum_{k\in\mathbb{Z}_{+}^{3}} \Big( \sum_{p\in L_{k}} 2 \lambda_{k,p} b_{k,p}^{\ast}b_{k,p} + \sum_{p\in L_{-k}} 2 \lambda_{-k,p} b_{-k,p}^{\ast}b_{-k,p} \Big). \end{align} $$

Combining (1.34) and (1.37), we arrive at a Hamiltonian quadratic in terms of the operators $b_{k,p}$ , which could be treated in the bosonic interpretation. Note that $b_{k,p}\psi _{\mathrm { FS}}=0$ for all $k\in \mathbb {Z}_{*}^3,p\in L_k$ , and hence, the Fermi state plays the role of the bosonic vacuum.

Overview of the heuristic assumptions behind the random phase approximation

In the physics literature [36, 37], the RPA entails two assumptions:

1. That the excitation operators $b_{k,p}^{\ast }$ , $b_{k,p}$ in (1.33) can be treated as bosonic creation and annihilation operators, and that the operators $b_{k,p}$ and $b_{l,q}$ with $k\neq l$ can be considered as acting on independent Fock spaces. Mathematically, we thus expect that the approximate canonical commutation relations (CCR)

(1.38) $$ \begin{align} \big[b_{k,p},b_{l,q}\big]=\big[b_{k,p}^{\ast},b_{l,q}^{\ast}\big]=0,\quad\big[b_{k,p},b_{l,q}^{\ast}\big]\approx\delta_{k,l}\delta_{p,q} \end{align} $$

should hold in an appropriate sense.

2. That the operator in (1.22) can be approximated by an effective Hamiltonian which is quadratic in terms of $b_{k,p}^{\ast }$ and $b_{k,p}$ . This is already true for the interaction part $\sum _{k\in \mathbb {Z}_{+}^{3}}H_{\text {int}}^{k}$ in (1.34), and in the RPA, the non-bosonizable terms

(1.39) $$ \begin{align} \frac{k_{F}^{-1}}{(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(\tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k}+\frac{1}{2}D_{k}^{\ast}D_{k}\right) \end{align} $$

are simply dropped. Moreover, the kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ is not exactly of the desired form, but it can be replaced by the right side of (1.37). All this leads to the effective Hamiltonian

(1.40) $$ \begin{align} \sum_{k\in\mathbb{Z}_{+}^{3}}H_{\text{Bog},k}=\sum_{k\in\mathbb{Z}_{+}^{3}}\left(2\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}+2\sum_{p\in L_{-k}}\lambda_{-k,p}b_{-k,p}^{\ast}b_{-k,p}+H_{\text{int}}^{k} - \frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right| \right) \end{align} $$

acting on the bosonic Fock space $\bigoplus _{k\in \mathbb {Z}_{+}^{3}}\mathcal {F}^{+}\left(\ell ^{2}\left(L_{k}\cup L_{-k}\right)\right)$ .

Consequently, since the operators $b_{k,p}$ and $b_{l,q}$ with $k\neq l$ are considered as acting independently, we can diagonalize separately each quadratic bosonic Hamiltonian $H_{\text {Bog},k}$ by a Bogolubov transformation $\mathcal {U}_{k}$ on $\mathcal {F}^{+}\left(\ell ^{2}\left(L_{k}\cup L_{-k}\right)\right)$ such that

(1.41) $$ \begin{align} \mathcal{U}_{k}H_{\text{Bog},k}\mathcal{U}_{k}^{\ast} =2\,\text{tr}\left(\widetilde{E}_{k}-h_{k}\right) - \frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right| +2\sum_{p\in L_{k}\cup L_{-k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,q}, \end{align} $$

where for every $k\in \mathbb {Z}_{\ast }^{3}$ , we denote the following quantities on $\ell ^{2}(L_{k})$ :

(1.42) $$ \begin{align} \widetilde{E}_{k}= (h_{k}^{\frac{1}{2}}\left(h_{k}+2P_{v_{k}}\right)h_{k}^{\frac{1}{2}})^{\frac{1}{2}}, \quad h_{k}e_{p}=\lambda_{k,p}e_{p}, \quad P_{v_k} =|v_{k} \rangle \langle v_k|, \quad v_{k}=\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sum_{p\in L_{k}}e_{p} \end{align} $$

with $(e_{p})_{p\in L_{k}}$ the standard orthonormal basis of $\ell ^{2}(L_{k})$ .

Summing over k, we obtain the correlation energy (see Proposition 7.1)

(1.43) $$ \begin{align} E_{\mathrm{corr}} = \sum_{k\in \mathbb{Z}^3_+} \left( \text{tr}\left(\widetilde{E}_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right| \right)=\sum_{k\in \mathbb{Z}^3_{*}} \frac{1}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt, \end{align} $$

where $F (x )=\log \left(1+x\right)-x$ . All in all, the RPA thus suggests that up to a unitary transformation, we expect that

(1.44) $$ \begin{align} H_{N} \approx E_{\mathrm{FS}} + E_{\mathrm{corr}} +2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,p}, \end{align} $$

at least on states with few excitations.

Prediction of the correlation energy and the excitation spectrum

Equation (1.44) leads immediately to the following approximation for the ground state energy

(1.45) $$ \begin{align} \inf\sigma\left(H_{N}\right)\approx E_{\mathrm{FS}} +E_{\mathrm{corr}}, \end{align} $$

which coincides with [37, Eq. (34)], ⁴ where the authors derived it from the effective operator of equation (1.40) and also explained the connection to the original work of Gell-Mann–Brueckner [20]. See also [35, Eq. (9.54)] and [18, Eq. (12.53)] for this expression of the ground state energy.

More importantly, (1.44) also suggests that the excitation spectrum of $H_{N}$ could be described in terms of the eigenvalues of $2\widetilde {E}_{k}$ , which correspond to the bosonic elementary excitations and can be explicitly computed.

Indeed, for every eigenvalue $\epsilon $ of $\widetilde {E}_{k}$ , we may find an eigenvector $w\in \ell ^{2}(L_{k})$ such that

(1.46) $$ \begin{align} \epsilon^{2}w=\widetilde{E}_{k}^2 w = h_{k}^{\frac{1}{2}}\left(h_{k}+2P_{v_{k}}\right)h_{k}^{\frac{1}{2}} w =h_{k}^{2}w+2\langle h_{k}^{\frac{1}{2}}v_{k},w\rangle h_{k}^{\frac{1}{2}}v_{k}. \end{align} $$

But either $\epsilon $ is also an eigenvalue of $h_{k}$ or $\epsilon ^{2}-h_{k}^{2}$ is invertible. In the latter case, we can write

(1.47) $$ \begin{align} w=2\langle h_{k}^{\frac{1}{2}}v_{k},w\rangle(\epsilon^{2}-h_{k}^{2})^{-1}h_{k}^{\frac{1}{2}}v_{k}, \end{align} $$

and taking the inner product with $h_{k}^{\frac {1}{2}}v_{k}$ and cancelling the factors of $\langle h_{k}^{\frac {1}{2}}v_{k},w\rangle $ yields

(1.48) $$ \begin{align} 1=2\langle v_{k},h_{k}(\epsilon^{2}-h_{k}^{2})^{-1}v_{k}\rangle=\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\epsilon^{2}-\lambda_{k,p}^{2}}, \end{align} $$

which appears in [37, Eq. (6)]. The sum can be rewritten as

(1.49) $$ \begin{align} 1=\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p\in B_{F}}\frac{|k|^{2}}{\left(\epsilon-k\cdot p\right)^{2}-\left(\frac{1}{2}|k|^2\right)^{2}}. \end{align} $$

The formula (1.49) allows to compute all eigenvalues of $\widetilde {E}_{k}$ outside the spectrum of $h_k$ .

In the physically relevant case of the Coulomb potential where $\hat {V}_{k}k_{F}^{-1}$ is replaced by $4\pi e^{2}|k|^{-2}$ , one can immediately derive the famous plasmon frequency from (1.49): for $|k|\ll k_F^{1/2}$ , the largest eigenvalue $\epsilon $ is proportional to $k_F^{3/2}$ (see [14, Eq. (2.27)–(2.54)] for a detailed explanation), and its leading order behavior can be computed easily in the thermodynamic limit (including also a factor of $2$ for the electron spin states)

(1.50) $$ \begin{align} \epsilon^{2}=\frac{4\pi e^{2}}{(2\pi)^{3}}\int_{\overline{B}(0,k_{F})}\frac{\epsilon^{2}}{\left(\epsilon-k\cdot p\right)^{2}-\left(\frac{1}{2}|k|^2\right)^{2}}\,dp\approx\frac{2e^{2}}{(2\pi)^{2}}\text{Vol}(\overline{B}(0,k_{F}))=\frac{2e^{2}}{3\pi}k_{F}^{3}=2\pi ne^{2}, \end{align} $$

where $n=\frac {N}{\mathcal {V}}=\frac {1}{3\pi ^{2}}k_{F}^{3}$ is the number density of the system. Recalling that the relevant operator is $2\widetilde {E}_{k}$ rather than $\widetilde {E}_{k}$ and that $\frac {\hbar ^{2}}{2m}=1$ , this yields an excitation energy of

(1.51) $$ \begin{align} 2\epsilon\approx2\sqrt{2\pi ne^{2}}=\hbar\sqrt{\frac{4\pi ne^{2}}{m}} = \hbar \omega_{\text{plasmon}}, \end{align} $$

where $\omega _{\text {plasmon}}=\sqrt { {4\pi ne^{2}}m^{-1}}$ is called the plasmon frequency in [33, Eq. (3-90)] and [18, Eq. (15.16) - (15.18)]. Note that the Coulomb potential is special as it makes the right-hand side of (1.51) independent of k. See also [4, 14] where (1.51) was discussed.

Establishing the above heuristic computation is a longstanding problem in mathematical physics. In the present paper, we will give a rigorous formulation for the operator approximation (1.44) and then use this to justify the prediction of the correlation energy and the bosonic elementary excitations for a wide class of bounded potentials in the mean-field regime.

1.3 Main results

Our first result is the following rigorous formulation of the operator approximation (1.44).

Theorem 1.1 (Operator formulation of the RPA).

Let $V:\mathbb {T}^{3}\rightarrow \mathbb {R}$ obey $\hat {V}_{k}\geq 0$ and $\hat {V}_{-k}=\hat {V}_{k}$ for all $k\in \mathbb {Z}^{3}$ , and assume furthermore that $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}|k|<\infty $ . Consider the Hamiltonian $H_N$ given in (1.1) with $N=|B_F|$ . Let the operators $H_{\mathrm {kin}}'$ , $\mathcal {N}_E$ , $\widetilde {E}_{k}-h_k$ be defined in (1.11), (1.13), (1.42). Let the energies $E_{\mathrm {FS}}$ , $E_{\mathrm {corr}}$ be defined in (1.22), (1.43). Then there exists a unitary transformation $\mathcal {U}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ such that

(1.52) $$ \begin{align} \mathcal{U}H_{N}\mathcal{U}^{\ast} = E_{\mathrm{FS}} +E_{\mathrm{corr}} +H_{\mathrm{eff}} +\mathcal{E}_{\mathcal{U}}, \end{align} $$

where the effective operator $H_{\text {eff}}: \mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ is

(1.53) $$ \begin{align} H_{\mathrm{eff}}=H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in\mathbb{Z}^{3}_{*} }\sum_{p,q\in L_{k}}\left\langle e_{p},(\widetilde{E}_{k}-h_{k})e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q} \end{align} $$

and the error operator $\mathcal {E}_{\mathcal {U}}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ obeys the operator inequality: for every constant $\epsilon>0$ ,

(1.54) $$ \begin{align} \pm\mathcal{E}_{\mathcal U}\leq Ck_F^{-\frac 1 {94}+\epsilon} \left(k_{F}^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime}+ k_F \right),\quad k_{F}\rightarrow\infty. \end{align} $$

The unitary operator in Theorem 1.1 is given explicitly as $\mathcal {U}=e^{\mathcal {J}}e^{\mathcal {K}}$ , where $\mathcal {K}$ and $\mathcal {J}$ are given in (1.78) and (1.85), respectively (the transformations $e^{\mathcal {K}}$ and $e^{\mathcal {J}}$ are studied in detail in Sections 5 and 9).

Remark 1.1. The operator $\mathcal {N}_E H_{\mathrm {kin}}'$ on the right-hand side of (1.54) is nothing but the ‘bosonic kinetic operator’, due to the following remarkable identity (see Proposition 10.1):

(1.55) $$ \begin{align} 2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}=\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

Thus, in Theorem 1.1, we control the error in the random phase approximation using only the fermionic and bosonic kinetic operators, which is very natural.

Remark 1.2. In the expansion (1.52), $E_{\mathrm {FS}}$ is of order $k_F^5$ , and $E_{\mathrm {corr}}$ is of order $k_F$ . As we will argue below, when we apply this to the low-lying eigenstates with energy $E_{\mathrm {FS}}+ O(k_F)$ , the expectation of the effective Hamiltonian $H_{\mathrm {eff}}$ in (1.53) is of order $k_F$ , while the error term $\mathcal {E}_{\mathcal {U}}$ in (1.54) is of order $O(k_F^{1-\frac 1 {94}+\epsilon })=o(k_F)$ .

In order to put Theorem 1.1 to good use, we need some a priori estimate on the low-lying eigenstates of the Hamiltonian $H_N$ . We have the following:

Theorem 1.2 (A priori estimate for eigenstates).

Let V and $\mathcal {U}$ be as in Theorem 1.1. Let $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ be a normalized eigenstate of $H_{N}$ with energy $\left\langle \Psi ,H_{N}\Psi \right\rangle \le E_{\mathrm {FS}} + \kappa k_F$ for some constant $\kappa>0$ independent of $k_F$ . Then,

$$\begin{align*}\left\langle \Psi, \left(H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right) \Psi\right\rangle \leq C (\kappa +1)^2 k_F \end{align*}$$

for a constant $C>0$ depending only on V. The same bound holds with $\Psi $ replaced by $\mathcal {U} \Psi $ .

Remark 1.3. Thanks to the inequality $\mathcal {N}_{E}\leq H_{\operatorname {\mathrm {kin}}}^{\prime }$ (see [6, Lemma 2.4] and also Proposition 2.1 below), Theorem 1.2 implies that for an eigenstate $\Psi $ of $H_{N}$ with energy $\left\langle \Psi ,H_{N}\Psi \right\rangle \le E_{\mathrm {FS}} + O(k_F)$ , we have

(1.56) $$ \begin{align} \left\langle \Psi,\mathcal{N}_{E} \Psi\right\rangle \le \left\langle \Psi, H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle = O(k_F). \end{align} $$

Thus, the number of excitations is much smaller than the total number of particles ( $k_F\sim N^{1/3}\ll N$ ). While (1.56) has been derived in [24, 6] for every state with energy $\left\langle \Psi ,H_{N}\Psi \right\rangle \le E_{\mathrm {FS}} + O(k_F)$ (at least for a class of potentials V), the improved bound in Theorem 1.2 is deeper, and the eigenstate assumption plays a crucial role in the proof.

From Theorems 1.1 and 1.2, we can deduce immediately the asymptotic formula (1.45) on the ground state energy up to an error $o(k_F)$ . Indeed, the energy upper bound is given by the trial state $\mathcal {U}^* \psi _{\mathrm {FS}}$ , while the energy lower bound follows from the obvious operator inequality $\widetilde {E}_{k} \ge h_k$ . Moreover, our approach is quantitative, and we can derive (1.45) with explicit error estimates.

Theorem 1.3 (Ground state energy).

Let V be as in Theorem 1.1. Then for all $\epsilon>0$ ,

$$\begin{align*}\inf\sigma\left(H_{N}\right)= E_{\mathrm{FS}} +E_{\mathrm{corr}} +O(k_{F}^{1-\frac{1}{94} +\epsilon}),\quad k_{F}\rightarrow\infty. \end{align*}$$

Here are some remarks concerning Theorem 1.3.

Remark 1.4. The method of our proof can be adapted to give the upper bound under the weaker condition $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}^{2}|k|<\infty $ (see [8, Appendix A] for a derivation of the upper bound under this weaker condition). Additionally, under this condition it can be shown that

(1.57) $$ \begin{align} \frac{1}{\pi}\sum_{k\in\mathbb{Z}^{3}_{*} }\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt=\frac{k_{F}}{\pi}\sum_{k\in\mathbb{Z}^{3}_{*} }|k|\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}}{(2\pi)^{2}}I (t )\right)dt+o\left(k_{F}\right), \end{align} $$

where $F (x )=\log \left(1+x\right)-x$ and $I (t )=1-t\tan ^{-1}\left(t^{-1}\right)$ (this essentially amounts to replacing the Riemann sum by the integral and can be done by following either the proof of [5, Eq. (5.15)] or the analysis in Appendix A; the condition $\sum \hat {V}_{k}^{2}|k|<\infty $ ensures that the main contribution comes from $|k|\sim O(1)$ ). Hence, Theorem 1.3 implies that

(1.58) $$ \begin{align} \inf\sigma\left(H_{N}\right)= E_{\mathrm{FS}} +\frac{k_{F}}{\pi}\sum_{k\in\mathbb{Z}^{3}_{*} }|k|\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}}{(2\pi)^{2}}I (t )\right)dt+o\left(k_{F}\right). \end{align} $$

A result similar to ours, namely, the bound (1.58) for all potentials satisfying $\sum _k \hat V_k |k|<\infty $ , has been independently obtained in [8], based on a refinement of the method in [5, 6]. ⁵ The bound (1.58) was proved earlier in [5, 6], under the additional assumption that the Fourier coefficients $\hat {V}_{k}$ be finitely supported and that $\Vert \hat {V}\Vert _{\ell ^{1}}$ be sufficiently small. For small $\hat {V}_{k}$ , the logarithm of equation (1.58) can be expanded for

(1.59) $$ \begin{align} \sigma\left(H_{N}\right)= E_{\mathrm{FS}} -\frac{1 - \log\left(2\right)}{6(2\pi)^{4}}k_{F}\sum_{k\in\mathbb{Z}^{3}_{*} }\hat{V}_{k}^{2}|k|\left(1+O\left(\hat{V}_{k}\right)\right)+o\left(k_{F}\right), \end{align} $$

which was first proved in [24].

Remark 1.5. A further refinement of our method allows a derivation of a rigorous energy upper bound for all potentials satisfying $\sum _{k\ne 0} \hat V_k^2<\infty $ ; see [15]. This covers the case of the Coulomb potential $\hat {V}_{k} = 4\pi e^2 |k|^{-2}$ , where the correlation energy is given by the left-hand side of (1.57) which is of order $k_F \log k_{F}$ plus a correlation exchange correction of order $k_F$ (the correlation exchange contribution comes from the fact that the purely bosonic picture is not exact; it is different from the exchange energy which is part of $E_{\mathrm {FS}}$ ). In particular, for the Coulomb potential, the right-hand side of (1.57) diverges, whereas the left-hand side does not, and hence the discrete form in (1.57) is arguably more fundamental than the continuous form. It is interesting that in our method the discrete version of the correlation energy always appears naturally.

Besides containing the information of the ground state energy, another decisive consequence of the operator statement in Theorem 1.1 is that it allows us to obtain all bosonic elementary excitations predicted in the physics literature. We have the following:

Theorem 1.4 (Bosonic elementary excitations).

Let V and $\mathcal {U}$ be as in Theorem 1.1. Let $\Psi \in \mathcal {H}_{N}$ be a normalized wave function such that $ \mathcal {N}_{E} \Psi = \Psi $ and $\langle \Psi , H_{\operatorname {\mathrm {kin}}}^{\prime } \Psi \rangle = O(k_F)$ . Then for all $\epsilon>0$ , we have

$$ \begin{align*} \langle \Psi, \mathcal{U}H_{N}\mathcal{U}^{\ast} \Psi\rangle = E_{\mathrm{FS}} + E_{\mathrm{corr}} + \langle \Psi, \left.H_{\mathrm{eff}}\right|_{\mathcal{N}_{E}=1} \Psi\rangle + O(k_{F}^{1-\frac{1}{94} +\epsilon}), \end{align*} $$

where

(1.60) $$ \begin{align} \left.H_{\mathrm{eff}}\right|_{\mathcal{N}_{E}=1}=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,q} = \tilde{U} \left( \bigoplus_{k\in\mathbb{Z}_{\ast}^{3}}2\widetilde{E}_{k} \right) \tilde{U}^{\ast} \end{align} $$

on the space $\left\{ \Psi \in \mathcal {H}_{N}\mid \mathcal {N}_{E}\Psi =\Psi \right\}$ , and

(1.61) $$ \begin{align} \tilde{U}:\bigoplus_{k\in\mathbb{Z}_{\ast}^{3}}L^{2}(L_{k})\rightarrow\left\{ \Psi\in\mathcal{H}_{N}\mid\mathcal{N}_{E}\Psi=\Psi\right\} \end{align} $$

is a unitary isomorphism defined by

(1.62) $$ \begin{align} \tilde{U}\bigoplus_{k\in\mathbb{Z}_{\ast}^{3}}\varphi_{k}=\sum_{k\in\mathbb{Z}_{\ast}^{3}}b_{k}^{\ast}\left(\varphi_{k}\right)\psi_{\mathrm{FS}} = \sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_k} \langle e_p, \varphi_k\rangle b_{k,p}^{\ast}\psi_{\mathrm{ FS}}. \end{align} $$

Recall that all eigenvalues of $\widetilde {E}_{k}$ can be computed explicitly from the spectrum of $h_k$ and (1.49). From Theorem 1.1 and Theorem 1.4, we may say that up to the unitary transformation $\mathcal {U}$ , the RPA is exact for the $\{\mathcal {N}_{E}=1\}$ eigenspace of the effective Hamiltonian $H_{\mathrm {eff}}$ . To our knowledge, this is the first rigorous derivation of the bosonic elementary excitations from first principles.

Remark 1.6. For every fixed $k\in \mathbb {Z}^3_{*}$ , in the limit $k_F\to \infty $ , most eigenvalues of $\widetilde {E}_{k}$ are of order $k_F$ , but the lowest eigenvalue of $\widetilde {E}_{k}$ is of order $o(k_F)$ . This absence of a one-body spectral gap corresponds to the expected fact that the excitation spectrum of $k_F^{-1}H_N$ becomes continuous in the limit $k_F\to \infty $ . Therefore, in principle, it is very difficult to extract useful information by analyzing the full spectrum of $H_N$ . The significance of Theorem 1.4 is to offer a nontrivial statement on the bosonic excitations by analyzing exactly the spectrum of the effective Hamiltonian instead of looking directly at the spectrum of $H_N$ .

Remark 1.7. In Theorem 1.4, the restriction to the $\mathcal {N}_{E}=1$ eigenspace is important. Obviously, the effective Hamiltonian (1.53) does not coincide with that in the heuristic formula (1.44). Hence, it is natural to ask what to make of the assumption of the RPA that the effective Hamiltonian should behave like a diagonalized bosonic Hamiltonian. To approach this question, we note that using (1.55), we can rewrite the effective Hamiltonian in (1.53) as

(1.63) $$ \begin{align} H_{\text{eff}}=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,q}-\left(\mathcal{N}_{E}-1\right)H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

Since this operator commutes with $\mathcal {N}_{E}$ , we can restrict $H_{\text {eff}}$ to the eigenspaces of $\mathcal {N}_{E}$ . Doing so, we see that the trivial eigenspace $\left\{ \mathcal {N}_{E}=0\right\} =\operatorname {\mathrm {span}}\left(\psi _{\mathrm { FS}}\right)$ exactly corresponds to the ground state energy which is already addressed in Theorem 1.3. For the first nontrivial eigenspace $\{\mathcal {N}_{E}=1\}$ , we do indeed obtain the expected operator

(1.64) $$ \begin{align} \left.H_{\text{eff}}\right|_{\mathcal{N}_{E}=1}=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,q}, \end{align} $$

as in the heuristic formula (1.44). Moreover, the second identity in (1.60) tells us that $\left.H_{\text {eff}}\right|_{\mathcal {N}_{E}=1}$ can be diagonalized explicitly on $\{\mathcal {N}_{E}=1\}$ , which is important for applications.

More generally, we can also consider the higher excitation sectors $\left\{ \mathcal {N}_{E}=M\right\} $ for $M\in \mathbb {N}$ .

Theorem 1.5 (Higher excitations).

Let V and $\mathcal {U}$ be as in Theorem 1.1. Let $1\le M\le O(k_F)$ . Let $\Psi \in \mathcal {H}_{N}$ be a normalized wave function such that $ \mathcal {N}_{E} \Psi = M\Psi $ and $\langle \Psi , H_{\operatorname {\mathrm {kin}}}^{\prime } \Psi \rangle \le O(k_F)$ . Then for all $\epsilon>0$ , we have

$$ \begin{align*} \langle \Psi, \mathcal{U}H_{N}\mathcal{U}^{\ast} \Psi\rangle = E_{\mathrm{FS}} + E_{\mathrm{corr}} + \langle \Psi, \left.H_{\mathrm{eff}}\right|_{\mathcal{N}_{E}=M} \Psi\rangle + O(k_{F}^{1-\frac{1}{94} +\epsilon}), \end{align*} $$

where

$$ \begin{align*} \left.H_{\mathrm{{eff}}}\right|_{\mathcal{N}_{E}=M}=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},(\widetilde{E}_{k}-(1-M^{-1})h_{k})e_{q}\rangle b_{k,p}^{\ast}b_{k,q}. \end{align*} $$

Remark 1.8. For $M\ge 2$ , the operator $\left.H_{\text {eff}}\right|_{\mathcal {N}_{E}=M}$ in Theorem 1.5 cannot be diagonalized explicitly as in (1.60). The quasi-bosonic property is insufficient to guarantee that it is diagonalizable, even approximately. Understanding the behaviour of $H_{\text {eff}}$ on higher eigenspaces and reconciling the RPA thus appears to be an interesting but nontrivial task. Some progress in this direction was done in [14] where the norm $\|(H_{\mathrm {{eff}}} - M \epsilon ) \Psi \|$ was estimated for suitable trial states.

1.4 Proof strategy

Now let us explain some key ingredients of the proof. Following [37], our approach consists of studying pair-excitations $b_{k,p}^{\ast }=c_{p}^{\ast }c_{p-k}$ , where $c_{p-k}$ annihilates a particle with momentum $p-k$ (i.e., creates a hole in the Fermi ball), and $c_{p}^{\ast }$ creates a particle outside the Fermi ball. These operators $b_{k,p}$ , $b_{k,p}^{\ast }$ satisfy the bosonic commutation relations in an appropriate sense. This enables the use of a quasi-bosonic Bogolubov transformation to diagonalize the original fermionic operator. A main achievement of the present work is the analytical elaboration of this bosonic picture.

In [5, 6], a different, collective bosonization approach was developed by averaging the pair-excitations $b_{k,p}^{\ast }$ on ‘patches’ near the surface of the Fermi ball, thus realizing strengthened versions of the bosonic commutation relations which make the comparison with the purely bosonic computation significantly easier. In the present paper, we show that the bosonization idea can be implemented directly for pairs of fermions without such an averaging procedure. In our opinion, this new approach is conceptually closer to the physics of the problem and more transparent for applications. In particular, it allows us to obtain all bosonic elementary excitations as in Theorem 1.4. Moreover, the new method is potentially applicable to Coulomb systems, where the correlation exchange correction to the purely bosonic computation plays an important role; see [15] for a rigorous ground state energy upper bound.

In the context of interacting Bose gases, Bogolubov transformations based on another approximate CCR have been used to study the excitation spectrum; see, for example, [38, 23, 9, 25]. However, for the fermionic problem considered in the present paper, the approximate CCR holds in a very different setting and requires distinct estimation techniques.

Now let us provide further details.

Bosonization method

The driving concept of the random phase approximation is the bosonization of fermionic pairs. We must therefore argue why the excitation operators

(1.65) $$ \begin{align} b_{k,p}=c_{p-k}^{\ast}c_{p},\quad b_{k,p}^{\ast}=c_{p}^{\ast}c_{p-k},\quad p\in L_{k}=\left(B_{F}+k\right)\backslash B_{F} \end{align} $$

obey an approximate CCR. Consider for simplicity the case $k=l$ : then computation shows that for any $p,q\in L_{k}$ , $[b_{k,p},b_{k,q}]=[b_{k,p}^{\ast },b_{k,q}^{\ast }]=0$ , but

(1.66) $$ \begin{align} [b_{k,p},b_{k,q}^{\ast}]=\delta_{p,q}-\delta_{p,q}(c_{p}^{\ast}c_{p}+c_{p-k}c_{p-k}^{\ast}). \end{align} $$

In general, thanks to Pauli’s exclusion principle ( $c_{p}^{\ast }c_{p},\,c_{p}c_{p}^{\ast }\leq 1$ ), the error term in (1.66) satisfies the simple bound $\delta _{p,q}(c_{p}^{\ast }c_{p}+c_{p-k}c_{p-k}^{\ast })\leq 2\delta _{p,q}$ , but this is even bigger than the leading term $\delta _{p,q}$ . The key observation is that although these errors terms can not be considered to be small individually, they are so on average. For instance,

(1.67) $$ \begin{align} \sum_{p,q\in L_{k}}\delta_{p,q}(c_{p}^{\ast}c_{p}+c_{p-k}c_{p-k}^{\ast})=\sum_{p\in L_{k}}c_{p}^{\ast}c_{p}+\sum_{p\in L_{k}}c_{p-k}c_{p-k}^{\ast}\leq2\,\mathcal{N}_{E}, \end{align} $$

where $\mathcal {N}_{E}$ is the ‘excitation number operator’ defined in (1.13). Thus, for states where the expectation value of $\mathcal {N}_{E}$ is much smaller than $\sum _{p,q\in L_{k}}\delta _{p,q}=\left|L_{k}\right| \sim \min \{|k| k_F^2,k_F^3\}$ , one may expect that the contribution of the non-bosonic error terms are also smaller than the leading bosonic behaviour. Justifying this idea rigorously is one of the main results of this paper.

Note that unlike the works [24, 5, 6], we do not employ the ‘particle-hole transformation’ R, which maps $\psi _{\mathrm {FS}}$ to the vacuum, so that we always work directly on the space $\mathcal {H}_{N}$ .

A priori estimates

As explained above, to apply the bosonization method, we need to show that the expectation of $\mathcal {N}_E$ against low-lying eigenstates of $H_N$ is much smaller than $\left|L_{k}\right|\sim \min \left\{ k_{F}^{2}|k|,k_{F}^{3}\right\}$ .

Using the condition $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}|k|<\infty $ and a variant of Onsager’s lemma, we can prove that

(1.68) $$ \begin{align} H_{N} \ge E_{\mathrm{FS}} + H_{\operatorname{\mathrm{kin}}}^{\prime} - Ck_{F}. \end{align} $$

Consequently, if $\Psi $ is any eigenstate for $H_N$ satisfying $\left\langle \Psi ,H_{N}\Psi \right\rangle \leq E_{\mathrm {FS}} +Ck_{F}$ , then

(1.69) $$ \begin{align} \left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \leq Ck_{F}. \end{align} $$

Since $H_{\operatorname {\mathrm {kin}}}^{\prime }\ge \mathcal {N}_{E}$ , which was already explained in [6], this implies that $\langle \Psi , \mathcal {N}_E \Psi \rangle \le Ck_F \ll |L_k|$ . For V sufficiently small, this bound was first proved in [24] (by a different method), and it was also used in [6]. In practice, we will also need a stronger a priori estimate, namely,

(1.70) $$ \begin{align} \left\langle \Psi, k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \leq Ck_{F} \end{align} $$

as stated in Theorem 1.2. This we will obtain by employing a bootstrapping argument for eigenstates, inspired by the ‘improved condensation’ in the context of Bose gases in [38, 23, 29, 30]. In [6], an analogue of equation (1.70) was proved for a modified ground state by using a ‘localization in Fock space’ technique. In comparison, our estimate of equation (1.70) is obtained in a far more direct fashion and yields a uniform bound for all low-lying eigenstates. In particular, thanks to (1.69) and (1.70), the operator estimate in Theorem 1.1 leads to direct consequences on the ground state energy and the excitation spectrum of $H_N$ .

Removing the non-bosonizable terms

An important ingredient of the RPA is that the non-bosonizable terms

(1.71) $$ \begin{align} \frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(2\,\mathrm{ Re}\left(\tilde{B}_{k}^{\ast}+\tilde{B}_{-k}\right)D_{k}+D_{k}^{\ast}D_{k}\right) \end{align} $$

are negligible to the leading order of the correlation energy. Here, we offer a direct estimate for these terms, which is simpler than the strategy proposed in [6] and does not require a smallness condition on V. More precisely, in Theorem 2.4, we will prove that the non-bosonizable terms are bounded by $o(1) (k_F^{-1}\mathcal {N}_E H_{\operatorname {\mathrm {kin}}}' + H_{\operatorname {\mathrm {kin}}}'+k_F)$ , and hence, the expectation against the low-lying eigenstates of $H_N$ is of order $o(k_F)$ due to the a priori estimates mentioned before.

Bosonization of the kinetic operator and the excitation number operator

Concerning the bosonizable terms, while the interaction terms can be interpreted directly as a quadratic Hamiltonian in the quasi-bosonic picture as in (1.34), the treatment of the kinetic operator is more subtle. In fact, (1.37) does not hold as a direct operator approximation. Instead, we will justify it by appealing to the commutator relation

(1.72) $$ \begin{align} \left[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{l,q}^{\ast}\right]\approx\left[2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p},b_{l,q}^{\ast}\right]. \end{align} $$

This commutator relation ensures that the difference

(1.73) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p} \end{align} $$

is essentially invariant under the Bogolubov transformations introduced later, which is sufficient for our purpose. The approximation (1.72) is a consequence of the exact commutation relation (1.35): For every $p\in L_k= B_F^c \cap (B_F+k)$ , by the CAR, we have

(1.74) $$ \begin{align} \left[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{k,p}^{\ast}\right] &= \sum_{q\in B_{F}^{c}} \left|q\right|^{2} \left[c_{q}^{\ast}c_{q},c_{p}^* c_{p-k} \right] - \sum_{q\in B_{F}}\left|q\right|^{2} \left[c_{q}c_{q}^{\ast}, c_{p}^* c_{p-k} \right] \nonumber\\ &=\sum_{q\in B_{F}^{c}} \left|q\right|^{2} \left[c_{q}^{\ast}c_{q},c_{p}^*\right] c_{p-k} - \sum_{q\in B_{F}}\left|q\right|^{2} c_{p}^* \left[c_{q}c_{q}^{\ast}, c_{p-k} \right] \nonumber \\ &=\sum_{q\in B_{F}^{c}} \left|q\right|^{2} \delta_{q,p} c_{q}^{\ast} c_{p-k} - \sum_{q\in B_{F}}\left|q\right|^{2} \delta_{q,p-k} c_{p}^* c_{p-k} \nonumber \\ & =|p|^{2} c_{p}^{\ast} c_{p-k} - |p-k|^{2} c_{p}^* c_{p-k} = \left(|p|^{2}-|p-k|^{2}\right)b_{k,p}^{\ast}. \end{align} $$

A similar strategy was used in [6], although the analysis there is more complicated due to the averaging technique of the ‘patches’. In particular, the operators on ‘patches’ in [6] do not obey the exact commutator relation $[H_{\operatorname {\mathrm {kin}}}^{\prime },b_{k,p}^{\ast }]=2\lambda _{k,p}b_{k,p}^{\ast }$ , and so the kinetic operator has to be handled by an additional linearization argument.

Note that in the same manner of the dispersion relation in (1.74), we also have

(1.75) $$ \begin{align} \big[\mathcal{N}_{E},b_{k,p}\big] & =\sum_{q\in B_{F}^{c}}\big[c_{q}^{\ast}c_{q},c_{p-k}^{\ast}c_{p}\big]=\sum_{q\in B_{F}^{c}}\big(c_{q}^{\ast}\big[c_{q},c_{p-k}^{\ast}c_{p}\big]+\big[c_{q}^{\ast},c_{p-k}^{\ast}c_{p}\big]c_{q}\big)\nonumber\\ & =\sum_{q\in B_{F}^{c}}\big(c_{q}^{\ast}\big(-c_{p-k}^{\ast}\big\{ c_{q},c_{p}\big\} +\big\{ c_{q},c_{p-k}^{\ast}\big\} c_{p}\big)+\big(-c_{p-k}^{\ast}\big\{ c_{q}^{\ast},c_{p}\big\} +\big\{ c_{q}^{\ast},c_{p-k}^{\ast}\big\} c_{p}\big)c_{q}\big)\nonumber \\ & =-\sum_{q\in B_{F}^{c}}\big(\delta_{q,p}c_{p-k}^{\ast}\big)c_{q}=-c_{p-k}^{\ast}c_{p}=-b_{k,p}\end{align} $$

for all $k\in \mathbb {Z}_{\ast }^{3}$ and $p\in L_{k}$ . This means that $\mathcal {N}_{E}$ plays the same role as the number operator in the bosonic picture.

Bogolubov transformation I

We will estimate the contribution of high momenta separately and only diagonalize the effective operator in (1.40) for low momenta. For this reason, we define a cutoff set

(1.76) $$ \begin{align} S_{C}=\overline{B}\left(0,k_{F}^{\gamma}\right)\cap\mathbb{Z}_{+}^{3}, \end{align} $$

where $\gamma \in (0,1]$ will be optimized later. For a given $k_{F}$ , we then diagonalize only

(1.77) $$ \begin{align} H_{\text{eff}}^{\prime}=\sum_{k\in S_{C}}\left(2\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}+2\sum_{p\in L_{-k}}\lambda_{-k,p}b_{-k,p}^{\ast}b_{-k,p}+H_{\text{int}}^{k}\right) \end{align} $$

and treat the remaining terms with $k\in \mathbb {Z}_{+}^{3}\backslash S_{C}$ as an error term. As $\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ forms an exhaustion of $\mathbb {Z}_{+}^{3}$ , all terms are thus nonetheless diagonalized in the limit $k_{F}\rightarrow \infty $ .

Inspired by the exact bosonic diagonalization (see Theorem 3.1 for details), we take the diagonalizing Bogolubov transformation to be of the form $e^{\mathcal {K}}$ for a generator $\mathcal {K}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ defined by

(1.78) $$ \begin{align} \mathcal{K}=\sum_{k\in S_{C}}\left(\sum_{p\in L_{k}}\sum_{q\in L_{-k}}\left\langle e_{p},K_{k}e_{-q}\right\rangle \left(b_{k,p}b_{-k,q}-b_{-k,q}^{\ast}b_{k,p}^{\ast}\right)\right), \end{align} $$

where the transformation kernels $K_{k}:\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ , $k\in \mathbb {Z}_{+}^{3}$ , are defined by

(1.79) $$ \begin{align} K_{k}=-\frac{1}{2}\log\left(h_{k}^{-\frac{1}{2}}\left(h_{k}^{\frac{1}{2}}\left(h_{k}+2P_{v_{k}}\right)h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}\right) \end{align} $$

with $h_{k},P_{v_{k}}$ as defined in equation (1.42). With this choice, we find that

(1.80) $$ \begin{align} e^{\mathcal{K}}H_{\text{eff}}^{\prime}e^{-\mathcal{K}}\approx\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\left(\text{tr}\left(E_{k}-h_{k}\right)+2\sum_{p,q\in L_{k}}\left\langle e_{p},E_{k}e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}\right) \end{align} $$

for

(1.81) $$ \begin{align} E_{k}=e^{-K_{k}}h_{k}e^{-K_{k}} \end{align} $$

and by the commutation relation of equation (1.72), that

(1.82) $$ \begin{align} e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}\right)e^{-\mathcal{K}}\approx H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p} \end{align} $$

so by the equations (1.77), (1.80) and (1.82), noting also that $\left\langle e_{p},h_{k}e_{q}\right\rangle =\delta _{p,q}\lambda _{k,p}$ ,

(1.83) $$ \begin{align} & e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}\cup\left(-S_{C}\right)}H_{\text{int}}^{k}\right)e^{-\mathcal{K}} \nonumber\\ &\quad \approx H_{\operatorname{\mathrm{kin}}}^{\prime} +\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\left(\text{tr}\left(E_{k}-h_{k}\right)+2\sum_{p,q\in L_{k}}\left\langle e_{p},\left(E_{k}-h_{k}\right)e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}\right).\end{align} $$

On the right side of (1.83), the constant $\sum _{k\in S_{C}\cup \left(-S_{C}\right)}\text {tr}\left(E_{k}-h_{k}\right)$ captures correctly the leading order of the correlation energy $E_{\mathrm {corr}}$ . However, although $E_{k}$ is isospectral to

(1.84) $$ \begin{align} \widetilde{E}_k=h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}=\left(h_{k}^{\frac{1}{2}}(h_{k}+2P_{v_{k}})h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}\geq h_{k}, \end{align} $$

the operator $E_{k}-h_{k}$ is not non-negative. Thus the term $2\sum _{p,q\in L_{k}}\left\langle e_{p},\left(E_{k}-h_{k}\right)e_{q}\right\rangle b_{k,p}^{\ast }b_{k,q}$ – a kind of second quantization of $E_{k}-h_{k}$ – cannot be ignored for the lower bound.

The Bogolubov transformation used in this part is analogous to that of [6]. It was proved in [6] that if V is small, then the quantization of $E_{k}-h_{k}$ can be controlled by $H_{\operatorname {\mathrm {kin}}}^{\prime }$ , leading to the desired lower bound on the ground state energy. In order to treat an arbitrary potential, we will instead utilize a second Bogolubov transformation which effectively replaces $E_k$ by $\widetilde {E}_k$ in (1.83).

Bogolubov transformation II

We define the second Bogolubov transformation $e^{\mathcal {J}}$ for a generator $\mathcal {J}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ defined by

(1.85) $$ \begin{align} \mathcal{J}=\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p,q\in L_{k}}\left\langle e_{p},J_{k}e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}, \end{align} $$

where $J_{k}=\log \left(U_{k}\right)$ denotes the (principal) logarithm of the unitary transformation $U_{k}:\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ defined by

(1.86) $$ \begin{align} U_{k}=\left(h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}e^{K_{k}}. \end{align} $$

This is precisely the unitary transformation which satisfies

(1.87) $$ \begin{align} U_{k}E_{k}U_{k}^{\ast}=h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}=\left(h_{k}^{\frac{1}{2}}(h_{k}+2P_{v_{k}})h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}} = \widetilde{E}_k, \end{align} $$

as is easily verified. This transformation acts such that

(1.88) $$ \begin{align} e^{\mathcal{J}}\left(\sum_{p,q\in L_{k}}\left\langle e_{p},E_{k}e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}\right)e^{-\mathcal{J}}\approx\sum_{p,q\in L_{k}}\left\langle e_{p},\widetilde{E}_k e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}, \end{align} $$

and thanks to the relation of equation (1.72), also

(1.89) $$ \begin{align} e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}\right)e^{-\mathcal{J}}\approx H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}, \end{align} $$

so all in all,

(1.90) $$ \begin{align} & e^{\mathcal{J}}e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}\cup\left(-S_{C}\right)}H_{\text{int}}^{k}\right)e^{-\mathcal{K}}e^{-\mathcal{J}} \nonumber\\ &\quad\approx\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\text{tr}\left(E_{k}-h_{k}\right)+H_{\operatorname{\mathrm{kin}}}^{\prime} +\sum_{k\in S_{C}\cup\left(-S_{C}\right)}\sum_{p,q\in L_{k}}\left\langle e_{p},(\widetilde{E}_k - h_k) e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}.\end{align} $$

As $\widetilde {E}_k-h_{k}\geq 0$ , the last term can now be dropped and the energy lower bound concluded. The cutoff $S_C$ can be removed at the end without serious difficulties. On the technical level, the second Bogolubov transformation is an important new tool to remove the smallness condition of [6], thus enabling us to work with a significantly larger class of interaction potentials. In the independent work [8], the idea of using the second Bogolubov transformation has also been introduced to refine the method in [5, 6].

Elementary excitations

The key ingredient to obtain all bosonic elementary excitations is the formula (1.60) in Theorem 1.4. To prove this, note that $\left.H_{\mathrm {eff}}\right|_{\mathcal {N}_{E}=1}$ commutes with $\mathcal {N}_{E}$ and the total momentum $P= \sum _{p\in \mathbb {Z}^3_{*}} p c_p^* c_p$ , so we may restrict $H_{\text {eff}}$ to the simultanous eigenspaces of $\mathcal {N}_{E}$ and P, which are

(1.91) $$ \begin{align} \left\{ \Psi\in\mathcal{H}_{N}\mid\mathcal{N}_{E}\Psi=\Psi,\,P\Psi=k\Psi\right\} =\operatorname{\mathrm{span}}(b_{k,p}^{\ast}\psi_{\mathrm{FS}})_{p\in L_{k}}=\left\{ b_{k}^{\ast}(\varphi )\psi_{\mathrm{FS}}\mid\varphi\in L^{2}(L_{k})\right\}. \end{align} $$

It turns out that the mapping $U_k:L^{2}(L_{k})\rightarrow \left\{ \Psi \in \mathcal {H}_{N}\mid \mathcal {N}_{E}\Psi =\Psi ,\,P\Psi =k\Psi \right\} $ defined by

(1.92) $$ \begin{align} U_k\varphi=b_{k}^{\ast}(\varphi )\psi_{\operatorname{\mathrm{FS}}},\quad\varphi\in L^{2}(L_{k}) \end{align} $$

is a unitary isomorphism with the property that

(1.93) $$ \begin{align} \left.H_{\text{eff}}\right|_{\mathcal{N}_{E}=1} = U_k (2\widetilde{E}_{k} ) U_k^{\ast}. \end{align} $$

Summing over different momenta k’s, we obtain the transformation $\tilde {U}$ introduced in (1.62).

In summary, our approach is different from the previous works [24, 5, 6] in many aspects. On the conceptual level, our direct bosonization method (i.e., working directly with the operators $b_{k,p}$ instead of averaging them on ‘patches’) allows us to stick closely to the heuristic argument of the physics literature and to obtain not only the ground state energy but also all bosonic elementary excitations, thus leading to the first complete justification of the RPA in the mean-field regime.

Although our general ideas are very transparent, to realize the whole procedure on a rigorous basis, we will need to develop several new estimates to justify all of the approximations made. In the rest of the paper, we will show how to implement the proof strategy rigorously.

Outline of the paper. In Section 2, we prove some general estimates involving the kinetic operator $H_{\operatorname {\mathrm {kin}}}$ and bound the non-bosonizable terms. In Section 3, we review the theory of bosonic Bogolubov transformations; in particular, we review how one may explicitly define a Bogolubov transformation which diagonalizes a given positive-definite quadratic Hamiltonian. We then apply the bosonic theory to our study of the Fermi gas where we implement the diagonalization procedure in the quasi-bosonic framework. This is done by introducing the quasi-bosonic quadratic Hamiltonian in Section 4 and the quasi-bosonic Bogolubov transformation $e^{\mathcal {K}}$ in Section 5 (these notations mirror the exact bosonic ones as closely as possible such that the bosonic theory is easily transferred to the quasi-bosonic setting). In this way, the quasi-bosonic analysis reduces to that of a collection of exact bosonic quadratic Hamiltonians plus correlation exchange terms – error terms which arise due to the deviation from the exact CCR. In Section 6, we estimate the exchange terms, reducing the analysis of these to the associated one-body operators of the bosonic problem. The one-body operators are studied separately in Section 7. In this part, we will need several estimates of Riemann sums, which are collected in the Appendix. We complete the analysis of the transformation $e^{\mathcal {K}}$ in Section 8, where we prove that $H_{\operatorname {\mathrm {kin}}}'$ and $\mathcal {N}_E$ are stable under the transformation $e^{\mathcal {K}}$ . In Section 9, we introduce the second unitary transformation $e^{\mathcal {J}}$ . The analysis of this transformation is essentially similar to the first one, except that we require new one-body operator estimates which are somewhat more difficult. Finally, we conclude the proofs of the main theorems in Section 10.

2 Removal of the non-bosonizable terms

In this section, we collect several basic estimates concerning the operator $H_N$ which can be obtained without using Bogolubov transformations. Recall the decomposition (1.22)

(2.1) $$ \begin{align} H_{N} =E_{\mathrm{FS}} +H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1}H_{\operatorname{\mathrm{int}}}^{\prime}, \quad E_{\mathrm{FS}} = \left\langle \psi_{\operatorname{\mathrm{FS}}},H_{N}\psi_{\operatorname{\mathrm{FS}}}\right\rangle. \end{align} $$

We will bound the interaction operator $H_{\operatorname {\mathrm {int}}}^{\prime }$ in terms of the kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ and then prove a priori estimates for eigenstates of $H_N$ which are parts of Theorem 1.2.

Recall the following result from [6, Lemma 2.4] concerning the kinetic operator $H_{\mathrm {kin}}'$ in (1.11).

Proposition 2.1. We have $H_{\operatorname {\mathrm {kin}}}^{\prime } \ge \mathcal {N}_{E}$ with $\mathcal {N}_{E}$ given in (1.13).

Proof. Since $|p|^2$ is an integer for $p\in \mathbb {Z}^3$ , our assumption $|B_F|=N$ implies that

(2.2) $$ \begin{align} \inf_{p\in B_{F}^{c}}|p|^{2} - \sup_{p\in B_{F}}|p|^{2} \ge 1. \end{align} $$

Therefore, in (1.14), we can choose $\zeta $ such that $\vert |p|^{2}-\zeta \vert \,\geq 1/2$ for all $p\in \mathbb {Z}^3$ .

Next, we consider the bosonizable terms in $H_{\operatorname {\mathrm {int}}}^{\prime }$ . The following result is a minor extension of [24, Lemma 4.7] (see also [6, Appendix B] for a simplified proof).

Proposition 2.2. For all $k\in \mathbb {Z}_{\ast }^{3}$ , the operator $\tilde {B}_{k}$ in (1.24) satisfies that

$$ \begin{align*} \tilde{B}_{k}^{\ast}\tilde{B}_{k}\leq Ck_{F}H_{\operatorname{\mathrm{kin}}}^{\prime}, \quad \tilde{B}_{k}\tilde{B}_{k}^{\ast} \leq Ck_{F} ( H_{\operatorname{\mathrm{kin}}}^{\prime} + |k| k_F), \end{align*} $$

where the constant $C>0$ is independent of k and $k_{F}$ .

Proof. As argued in [24, 6], for any $\Psi \in \mathcal {H}_{N}$ , it follows from the triangle and Cauchy-Schwarz inequalities that

(2.3) $$ \begin{align} \left\Vert \tilde{B}_{k}\Psi\right\Vert =\left\Vert \sum_{p\in L_{k}}c_{p-k}^{\ast}c_{p}\Psi\right\Vert \leq\sum_{p\in L_{k}}\left\Vert c_{p-k}^{\ast}c_{p}\Psi\right\Vert \leq\sqrt{\sum_{p\in L_{k}}\lambda_{k,p}^{-1}}\sqrt{\sum_{p\in L_{k}}\lambda_{k,p}\left\Vert c_{p-k}^{\ast}c_{p}\Psi\right\Vert ^{2}}, \end{align} $$

where $\lambda _{k,p}=\frac {1}{2}(|p|^{2}-|p-k|^{2})$ . Using (1.14) and Pauli’s exclusion principle $\|c_p\|_{\mathrm {op}}\le 1$ , $\|c^*_p\|_{\mathrm {op}}\le 1$ , we find that

(2.4) $$ \begin{align} \sum_{p\in L_{k}}\lambda_{k,p}\left\Vert c_{p-k}^{\ast}c_{p}\Psi\right\Vert ^{2} & =\frac{1}{2}\sum_{p\in L_{k}}\left(\vert|p|^{2}-\zeta\vert+\vert|p-k|^{2}-\zeta\vert\right)\left\Vert c_{p-k}^{\ast}c_{p}\Psi\right\Vert ^{2}\\ & \leq\frac{1}{2}\sum_{p\in L_{k}}\vert|p|^{2}-\zeta\vert\left\Vert c_{p}\Psi\right\Vert ^{2}+\frac{1}{2}\sum_{p\in L_{k}}\vert|p-k|^{2}-\zeta\vert\left\Vert c_{p-k}^{\ast}\Psi\right\Vert ^{2}\nonumber \\ & \leq \frac{1}{2}\sum_{p\in B_F^c}\vert|p|^{2}-\zeta\vert\left\Vert c_{p}\Psi\right\Vert ^{2}+\frac{1}{2}\sum_{p\in B_F}\vert|p|^{2}-\zeta\vert\left\Vert c_{p}^{\ast}\Psi\right\Vert ^{2}\nonumber =\frac{1}{2}\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle. \nonumber \end{align} $$

Thus, it remains to show that $\sum _{p\in L_{k}}\lambda _{k,p}^{-1}\leq Ck_{F}$ . For $|k|\sim O(1)$ , this bound was already proved in [24, 6]. For completeness, we will establish this bound for all $k\in \mathbb {Z}^3_{*}$ in the Appendix (Proposition A.2). Thus, in summary,

(2.5) $$ \begin{align} \tilde{B}_{k}^{\ast}\tilde{B}_{k}\leq\frac{1}{2}\left(\sum_{p\in L_{k}}\lambda_{k,p}^{-1}\right)H_{\operatorname{\mathrm{kin}}}^{\prime} \le Ck_{F}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

Then the bound for $\tilde {B}_{k}\tilde {B}_{k}^{\ast }$ follows from the fact that

(2.6) $$ \begin{align} \left[\tilde{B}_{k},\tilde{B}_{k}^{\ast}\right]=\left|L_{k}\right|-\sum_{p\in L_{k}}c_{p}^{\ast}c_{p}-\sum_{p\in L_{k}}c_{p-k}c_{p-k}^{\ast} \le |L_k| \le C|k| k_F^2. \end{align} $$

In the last estimate, we used $\left|L_{k}\right|\leq Ck_{F}^{2}|k|$ for all $k\in \mathbb {Z}_{\ast }^{3}$ (see Proposition A.1 for details).

For the non-bosonizable terms in $H_{\operatorname {\mathrm {int}}}^{\prime }$ , it was proved in [5, Eq. (5.1)] that

(2.7) $$ \begin{align} D_{k}^{\ast}D_{k}\leq4\,\mathcal{N}_{E}^{2}. \end{align} $$

However, this bound is not optimal for low-lying eigenfunctions (for which $\mathcal {N}_{E}\sim k_F$ ). In order to remove the non-bosonizable terms completely, we need the following improvement.

Proposition 2.3. For all $k\in \mathbb {Z}_{\ast }^{3}$ and any $0<\lambda \le \frac {1}{6}k_F^2$ , the operator $D_k$ in (1.23) satisfies

$$\begin{align*}D_{k}^{\ast}D_{k}\leq C\left(|k|^{-1}\lambda +|k|^{3+\frac{2}{3}}(\log k_{F})^{\frac{2}{3}}k_{F}^{\frac{2}{3}}\right)(\lambda +|k|) \mathcal{N}_{E}+C\lambda^{-\frac{1}{2}}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} \end{align*}$$

for a constant $C>0$ independent of k, $k_{F}$ and $\lambda $ .

In applications, we will eventually choose $\lambda =k_F^{2\gamma }/|k|^{4}$ for some constant $\gamma \in (0,1/9)$ .

Proof. For $k\in \mathbb {Z}_{\ast }^{3}$ , we write $D_{k}=D_{k}^{1}+D_{k}^{2}$ as in (1.24), namely,

(2.8) $$ \begin{align} D_{k}^{1}=\sum_{q\in B_{F}\cap\left(B_{F}+k\right)}c_{q-k}^{\ast}c_{q},\quad D_{k}^{2}= \sum_{q\in B_{F}^c \cap\left(B_{F}^c+k\right)}c_{q-k}^{\ast}c_{q}. \end{align} $$

By the Cauchy–Schwarz inequality,

(2.9) $$ \begin{align} D_{k}^{\ast}D_{k}\leq2\left((D_{k}^{1})^{\ast}D_{k}^{1}+(D_{k}^{2})^{\ast}D_{k}^{2}\right). \end{align} $$

We will estimate $(D_{k}^{1})^{\ast }D_{k}^{1}$ in detail, with the estimate of $(D_{k}^{2})^{\ast }D_{k}^{2}$ being similar. We have

(2.10) $$ \begin{align} (D_{k}^{1})^{\ast}D_{k}^{1} &=\sum_{p,q\in B_{F}\cap\left(B_{F}+k\right)} c_p^* c_{p-k} c_{q-k}^{\ast} c_{q} =\sum_{p,q \in B_{F}\cap\left(B_{F}+k\right)} ( \delta_{p,q}c_{p-k}c_{p-k}^{\ast} - c_{p-k}c_{q}c_{p}^{\ast}c_{q-k}^{\ast} ) \nonumber \\ &= \sum_{p \in B_{F}\cap\left(B_{F}+k\right)} c_{p-k}c_{p-k}^{\ast} - \frac{1}{2}\sum_{p,q\in B_{F}\cap\left(B_{F}+k\right)} ( c_{p-k}c_{q}c_{p}^{\ast}c_{q-k}^{\ast} + h.c.). \end{align} $$

Here, we used $k\ne 0$ so that $c_{p-k}$ and $c^*_p$ anti-commute. By the definition of $\mathcal {N}_E$ in (1.13),

(2.11) $$ \begin{align} \sum_{p\in B_{F}\cap\left(B_{F}+k\right)} c_{p-k}c_{p-k}^{\ast} \le \sum_{p\in B_{F}} c_{p}c_{p}^{\ast} = \mathcal{N}_E. \end{align} $$

Moreover, by the Cauchy–Schwarz inequality, for all $\epsilon _p>0$ , we get

(2.12) $$ \begin{align} &\pm \frac{1}{2}\sum_{p,q \in B_{F}\cap\left(B_{F}+k\right)} ( c_{p-k}c_{q}c_{p}^{\ast}c_{q-k}^{\ast} + h.c. ) \nonumber\\ & \le \frac{1}{2} \sum_{p,q \in B_{F}\cap\left(B_{F}+k\right)} (\epsilon_p c_{p-k}c_{q} c_q^* c_{p-k}^* + \epsilon_p^{-1} c_{q-k} c_p c_{p}^{\ast}c_{q-k}^{\ast}) \nonumber\\ & \le \frac{1}{2} \sum_{p,q \in B_{F}\cap\left(B_{F}+k\right)} ( \epsilon_p c_{p-k}c_{p-k}^* c_{q} c_q^* + \epsilon_p^{-1} c_{q-k} c_{q-k}^{\ast} c_p c_{p}^{\ast}) \nonumber\\ &\le \frac{1}{2} \sum_{p \in B_{F}\cap\left(B_{F}+k\right)} ( \epsilon_p c_{p-k}c_{p-k}^* + \epsilon_p^{-1}c_p c_{p}^{\ast} ) \mathcal{N}_E . \end{align} $$

By taking $\epsilon _p\equiv 1$ , we obtain immediately $(D_{k}^{1})^{\ast }D_{k}^{1} \le \mathcal {N}_E^2$ , which, together with a similar bound for $D_{k}^{2}$ , leads to (2.7). To improve on this, we have to choose $\epsilon _p$ differently.

Recall that in (1.14) we can choose $\zeta \in [\sup _{p\in B_{F}}|p|^{2},\inf _{p\in B_{F}^{c}}|p|^{2}]$ such that $\vert |p|^{2}-\zeta \vert \,\geq 1/2$ for all $p\in \mathbb {Z}^3$ . For any $\lambda>0$ , we can split

(2.13) $$ \begin{align} B_{F}\cap\left(B_{F}+k\right)=S_{k,\lambda}^{1}\cup S_{k,\geq\lambda}^{1}, \end{align} $$

where

(2.14) $$ \begin{align} S_{k,\lambda}^{1} &=\left\{ p\in B_{F}\cap\left(B_{F}+k\right)\mid \max\left\{ \vert|p|^{2}-\zeta\vert,\vert|p-k|^{2}-\zeta\vert\right\} <\lambda\right\} ,\nonumber\\ S_{k,\geq\lambda}^{1} &=\left\{ p\in B_{F}\cap\left(B_{F}+k\right)\mid\max\left\{ \vert|p|^{2}-\zeta\vert,\vert|p-k|^{2}-\zeta\vert\right\} \geq\lambda\right\} . \end{align} $$

Choosing $\epsilon _p=1$ for $p\in S_{k,\lambda }^{1}$ and using $\|c_p^*\|_{\mathrm {op}} \le 1$ , we get

(2.15) $$ \begin{align} \frac 1 2\sum_{p \in S_{k,\lambda}^{1}} ( \epsilon_p c_{p-k}c_{p-k}^* + \epsilon_p^{-1}c_p c_{p}^{\ast} ) \le |S_{k,\lambda}^{1}|. \end{align} $$

Choosing $\epsilon _p = \sqrt { | |p-k|^{2}-\zeta |} / \sqrt { | |p|^2 -\zeta |} $ for $p\in S_{k,\geq \lambda }^{1}$ , we have

(2.16) $$ \begin{align} & \sum_{p \in S_{k,\geq\lambda}^{1}} ( \epsilon_p c_{p-k}c_{p-k}^* + \epsilon_p^{-1}c_p c_{p}^{\ast} ) \nonumber\\ &\quad = \sum_{p \in S_{k,\geq\lambda}^{1}} \frac{1}{ \sqrt{| |p|^2 -\zeta| \cdot | |p-k|^{2}-\zeta| }}( | |p-k|^{2}-\zeta| c_{p-k}c_{p-k}^* + | |p|^2 -\zeta| c_p c_{p}^{\ast} ) \nonumber\\ &\quad \le \sum_{p \in S_{k,\geq\lambda}^{1}} \frac{1}{ \sqrt{\lambda/2} }( | |p-k|^{2}-\zeta| c_{p-k}c_{p-k}^* + | |p|^2 -\zeta| c_p c_{p}^{\ast} ) \le \frac{2\sqrt{2}}{\sqrt{\lambda}} H_{\mathrm{kin}}'. \end{align} $$

Here, we used that among two factors $| |p|^2 -\zeta |$ and $| |p-k|^{2}-\zeta | $ , there is at least one $\ge \lambda $ due to the assumption $p\in S_{k,\geq \lambda }^{1}$ , and the other one is trivially $\ge 1/2$ . In summary,

(2.17) $$ \begin{align} (D_{k}^{1})^{\ast}D_{k}^{1} \leq|S_{k,\lambda}^{1}| \mathcal{N}_{E} +C\lambda^{-\frac{1}{2}} H_{\operatorname{\mathrm{kin}}}^{\prime}\mathcal{N}_{E}. \end{align} $$

Similarly, we have

(2.18) $$ \begin{align} (D_{k}^{2})^{\ast}D_{k}^{2} \leq|S_{k,\lambda}^{2}| \mathcal{N}_{E} +C\lambda^{-\frac{1}{2}} H_{\operatorname{\mathrm{kin}}}^{\prime}\mathcal{N}_{E},\end{align} $$

where

(2.19) $$ \begin{align} S_{k,\lambda}^{2} =\left\{ p\in B_{F}^{c}\cap\left(B_{F}^{c}+k\right)\mid\max\left\{ \vert|p|^{2}-\zeta\vert,\vert|p-k|^{2}-\zeta\vert\right\} <\lambda\right\}. \end{align} $$

The desired conclusion of $D_k^* D_k$ follows from the bound

(2.20) $$ \begin{align} |S_{k,\lambda}^{1}|+|S_{k,\lambda}^{2}|\leq C\left(|k|^{-1}\lambda +|k|^{3+\frac{2}{3}}(\log k_{F})^{\frac{2}{3}}k_{F}^{\frac{2}{3}}\right)\left(\lambda +|k|\right) \end{align} $$

whose proof can be found in Proposition A.4 in the Appendix.

2.1 Estimation of the non-bosonizable terms

Now we are ready to remove the non-bosonizable terms, namely, the terms involving operators $D_k$ in the decomposition (1.28) of the interaction operator:

(2.21) $$ \begin{align} k_{F}^{-1} H_{\text{int}}^{\prime} = \sum_{k\in\mathbb{Z}_{+}^{3}}\left(H_{\text{int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right) +\frac{k_{F}^{-1}}{(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(\tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k}+\frac{1}{2}D_{k}^{\ast}D_{k}\right), \end{align} $$

where $H_{\text {int}}^{k}$ is defined in (1.29). Moreover, for technical reasons, we will also impose a momentum cutoff in the bosonizable terms. Recall the set $S_C$ in (1.76). Define

(2.22) $$ \begin{align} \mathcal{E}_{\mathrm{NB}} = k_{F}^{-1} H_{\text{int}}^{\prime} - \sum_{k\in S_C} \left(H_{\mathrm{ int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right). \end{align} $$

Proposition 2.4. Let $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}|k|<\infty $ . Then for all $\gamma \in (0,1/9)$ in $S_C$ , we have

$$ \begin{align*} \pm \mathcal{E}_{\mathrm{NB}} \le C k_F^{-\gamma/2} \Big(H_{\mathrm{kin}}' + k_F^{-1} \mathcal{N}_E H_{\mathrm{kin}}' + k_F\Big). \end{align*} $$

Here, the constant $C>0$ depends only on V (in particular, it is independent of $k,k_F$ and $\lambda $ ).

We write $\pm X\le Y$ for two operator inequalities $X\le Y$ and $-X\le Y$ .

Proof. For the bosonizable terms, by (2.6), Proposition 2.2 and Proposition 2.1, we can bound

(2.23) $$ \begin{align} \pm ( \{\tilde{B}_{k}^{\ast},\tilde{B}_{k}\} - |L_k| ) = \pm \left( 2 \tilde{B}_{k}^{\ast}\tilde{B}_{k} - \sum_{p\in L_{k}}c_{p}^{\ast}c_{p}-\sum_{p\in L_{k}}c_{p-k}c_{p-k}^{\ast} \right) \le 2 \tilde{B}_{k}^{\ast}\tilde{B}_{k} + \mathcal{N}_E\le C k_F H_{\mathrm{kin}}' \end{align} $$

for all $k\in \mathbb {Z}^3_{*}$ . Moreover, by the Cauchy–Schwarz inequality,

(2.24) $$ \begin{align} \pm (\tilde{B}_{k}^* \tilde{B}_{-k}^* + \tilde{B}_{-k}\tilde{B}_{k} ) \le |k|^{-1/2} \tilde{B}_{-k} \tilde{B}_{-k}^* + |k|^{1/2} \tilde{B}_{k}^* \tilde{B}_{k} \le C |k|^{1/2} k_F (H_{\mathrm{kin}}' + k_F) \end{align} $$

for all $k\in \mathbb {Z}^3_{*}$ . Combining (2.23) and (2.24), we find that

(2.25) $$ \begin{align} \pm \sum_{k \in \mathbb{Z}_{+}^{3} \backslash S_C} \left(H_{\mathrm{ int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right) &\le C (H_{\mathrm{kin}}' + k_F) \sum_{k \in \mathbb{Z}_{+}^{3} \backslash S_C} \hat V_k |k|^{1/2}\nonumber\\ & \le C (H_{\mathrm{kin}}' + k_F) k_F^{-\gamma/2} \sum_{k \in \mathbb{Z}^{3}} \hat V_k |k|. \end{align} $$

For the non-bosonizable terms, by the Cauchy–Schwarz inequality and Proposition 2.2, we have

(2.26) $$ \begin{align} \pm \left( \tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k} \right) \le k_F^{-\gamma/2} |k| \tilde{B}_{k}^{\ast} \tilde{B}_{k} + k_F^{\gamma/2} |k|^{-1} D_{k}^{\ast} D_{k} \le C k_F^{1-\gamma/2} |k| H_{\mathrm{kin}}' + k_F^{\gamma/2} |k|^{-1} D_{k}^{\ast} D_{k}, \end{align} $$

and hence,

(2.27) $$ \begin{align} \pm \sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k} \left( \tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k} + D_{k}^{\ast} D_{k} \right) \le C k_F^{1-\gamma/2} H_{\mathrm{kin}}' + \sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k} ( k_F^{\gamma/2} |k|^{-1} +1 ) D_{k}^{\ast} D_{k}. \end{align} $$

Let us decompose the sum on the right-hand side of (2.27) into the high-momenta $|k|> k_F^{\gamma /2}$ and the low-momenta $|k|\le k_F^{\gamma /2}$ . For the high-momenta, from the simple bound (2.7), we get

(2.28) $$ \begin{align} \sum_{k\in\mathbb{Z}_{\ast}^{3}, |k|> k_F^{\gamma/2}}\hat{V}_{k} ( k_F^{\gamma/2} |k|^{-1} +1 ) D_{k}^{\ast} D_{k} \le C k_F^{-\gamma/2} \mathcal{N}_E^2 \sum_{k\in\mathbb{Z}^{3}}\hat{V}_{k} |k|. \end{align} $$

For the low-momenta, using Proposition 2.3 with $\lambda =k_F^{\gamma }/|k|^{2}$ , we have

(2.29) $$ \begin{align} D_{k}^{\ast}D_{k} \le C\left( k_F^{2\gamma} + k_F^\gamma |k|^{2/3} (\log k_F)^{\frac 2 3} k_F^{2/3} + |k|^{3+2/3} (\log k_F)^{\frac 2 3} k_F^{2/3}\right) |k|\mathcal{N}_E +Ck_F^{-\gamma/2} |k| \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}, \end{align} $$

and hence,

(2.30) $$ \begin{align} \sum_{k\in\mathbb{Z}_{\ast}^{3}, |k|\le k_F^{\gamma/2}}\hat{V}_{k} D_{k}^{\ast}D_{k} &\le C \left( \sum_{k\in\mathbb{Z}^{3}}\hat{V}_{k} |k| \right) \left( k_F^{(3+\frac 2 3) \frac \gamma 2+ \frac 2 3} (\log k_F)^{\frac 2 3} \mathcal{N}_E + k_F^{-\frac \gamma 2} \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right) \nonumber\\ &\le C k_F^{-\gamma/2} \left( k_F \mathcal{N}_E +\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right) \end{align} $$

for all $\gamma \in (0,1/7)$ . Moreover, using Proposition 2.3 with $\lambda =k_F^{2\gamma }/|k|^{4}$ , we have

(2.31) $$ \begin{align} k_F^{\gamma/2} |k|^{-1} D_{k}^{\ast}D_{k} &\leq C\left( k_F^{4\gamma} + k_F^{(2+\frac 1 2)\gamma} (\log k_F)^{\frac 2 3} k_F^{2/3} + k_F^{\frac \gamma 2}|k|^{2+2/3} (\log k_F)^{\frac 2 3} k_F^{2/3}\right) |k|\mathcal{N}_E \nonumber\\ &\qquad +Ck_F^{-\gamma/2} |k| \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}, \end{align} $$

and hence,

(2.32) $$ \begin{align} \sum_{k\in\mathbb{Z}_{\ast}^{3}, |k|\le k_F^{\gamma/2}}\hat{V}_{k} k_F^{\gamma/2} |k|^{-1} D_{k}^{\ast}D_{k} &\le C \left( \sum_{k\in\mathbb{Z}^{3}}\hat{V}_{k} |k| \right) \left( k_F^{(2+\frac 1 2) \gamma + \frac 2 3} (\log k_F)^{\frac 2 3} \mathcal{N}_E + k_F^{-\frac \gamma 2} \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right) \nonumber\\ &\le C k_F^{-\gamma/2} \left( k_F \mathcal{N}_E +\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right) \end{align} $$

for all $\gamma \in (0,1/9)$ . Inserting (2.28), (2.30) and (2.32) in (2.27) and using Proposition 2.1, we conclude that

(2.33) $$ \begin{align} \pm \frac{k_{F}^{-1}}{(2\pi)^{3}} \sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k} \left( \tilde{B}_{k}^{\ast}D_{k}+D_{k}^{\ast}\tilde{B}_{k} + D_{k}^{\ast} D_{k} \right) \le C k_F^{-\gamma/2} \Big( H_{\mathrm{kin}}' + k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} \Big) \end{align} $$

for all $\gamma \in (0,1/9)$ . The conclusion follows from (2.25) and (2.33).

3 Overview of bosonic Bogolubov transformations

In this section, we review the general theory of quadratic Hamiltonians and Bogolubov transformations in the exact bosonic setting. Later, in the remainder of the paper, the analysis here will be adapted to handle the quasi-bosonic case where error terms have to be estimated carefully.

The study of bosonic quadratic Hamiltonians goes back to Bogolubov’s 1947 paper [10] where he proposed an effective Hamiltonian to describe the excitation spectrum of weakly interacting Bose gases. An important property of quadratic Hamiltonians is that they can be diagonalized by suitable Bogolubov transformations; see, for example, [2, 31, 16] for recent results in the infinite dimensional cases. For our application, we will only focus on the situation where the one-body Hilbert space is real and finite dimensional. Historically, the diagonalization problem in finite dimensions can be solved abstractly by using Williamson’s theorem [43]. We refer to [27] and [16, Section 2] for systematic discussions on the finite dimensional case.

In the present paper, we will need an explicit construction of the diagonalizing transformations so that we can adapt this to the quasi-bosonic operators. Such an explicit construction can be found in [23], which was also used in the fermionic context in [5, 6] and will be recalled below. Here, we will offer a slightly different treatment of Bogolubov transformations, in that we will view quadratic operators on Fock spaces as the fundamental object of study rather than the creation and annihilation operators.

Notation. We will denote by V a finite-dimensional real Hilbert space and let $n=\dim \left(V\right)$ . The bosonic Fock space associated to V is

(3.1) $$ \begin{align} \mathcal{F}^{+}\left(V\right)=\bigoplus_{N=0}^{\infty}\bigotimes_{\text{Sym}}^{N}V, \end{align} $$

where $\bigotimes _{\text {Sym}}^{N}V$ denotes the space of symmetric N-fold tensor products of V. To any element $\varphi \in V$ , there are associated two operators on $\mathcal {F}^{+}\left(V\right)$ : the annihilation operator $a(\varphi )$ and the creation operator $a^{\ast }(\varphi )$ . These are (formal) adjoints of one another and obey the canonical commutation relations (CCR): for any $\varphi ,\psi \in V$ ,

(3.2) $$ \begin{align} \left[a(\varphi ),a\left(\psi\right)\right] =\left[a^{\ast}(\varphi ),a^{\ast}\left(\psi\right)\right]=0, \quad \left[a(\varphi ),a^{\ast}\left(\psi\right)\right] =\left\langle \varphi,\psi\right\rangle. \end{align} $$

Additionally, the mappings $\varphi \mapsto a(\varphi )$ , $\varphi \mapsto a^{\ast }(\varphi )$ are linear. ⁶

3.1 Quadratic Hamiltonians

Similarly to how we can to any $\varphi \in V$ associate the two operators $a(\varphi )$ and $a^{\ast }(\varphi )$ , we may also associate two types of symmetric operators on $\mathcal {F}^{+}\left(V\right)$ to any symmetric operator on V. For the definition, we let $(e_{i})_{i=1}^{n}$ denote an orthonormal basis of V. Given any symmetric operator $A:V\rightarrow V$ , we then define the operator $Q_{1}(A)$ on $\mathcal {F}^{+}\left(V\right)$ by

(3.3) $$ \begin{align} Q_{1}(A)=\sum_{i,j=1}^{n}\left\langle e_{i},Ae_{j}\right\rangle \left(a^{\ast}(e_{i})a(e_{j})+a(e_{j})a^{\ast}(e_{i})\right), \end{align} $$

and likewise, for any symmetric operator $B:V\rightarrow V$ , we define the operator $Q_{2}(B)$ by

(3.4) $$ \begin{align} Q_{2}(B)=\sum_{i,j=1}^{n}\left\langle e_{i},Be_{j}\right\rangle \left(a^{\ast}(e_{i})a^{\ast}(e_{j})+a(e_{j})a(e_{i})\right). \end{align} $$

These definitions are independent of the basis chosen, and we can write equivalently

(3.5) $$ \begin{align} Q_{1}(A) & =\sum_{i=1}^{n}\left(a^{\ast}(Ae_{i})a(e_{i})+a(e_{i})a^{\ast}(Ae_{i})\right)\\ Q_{2}(B) & =\sum_{i=1}^{n}\left(a^{\ast}\left(Be_{i}\right)a^{\ast}(e_{i})+a(e_{i})a\left(Be_{i}\right)\right).\nonumber \end{align} $$

Thus, for real, symmetric $A,B:V\rightarrow V$ , we can define a quadratic Hamiltonian on $\mathcal {F}^{+}\left(V\right)$ by

(3.6) $$ \begin{align} H=Q_{1}(A)+Q_{2}(B). \end{align} $$

Note that by the CCR, we may express $Q_{1}(A)$ as

(3.7) $$ \begin{align} Q_{1}(A) =2\sum_{i,j=1}^{n}\left\langle e_{i},Ae_{j}\right\rangle a^{\ast}(e_{i})a(e_{j})+\text{tr}(A)=2\,\text{d}\Gamma(A)+\text{tr}(A), \end{align} $$

where $\text {d}\Gamma (A)$ denotes the second quantization of $A:V\rightarrow V$ . Sometimes in the literature, in particular in infinite dimensions, quadratic Hamiltonians are defined by $\text {d}\Gamma (A)+Q_{2}(B)$ , which is the same to our definition up to the constant $\text {tr}(A)$ . Here, we prefer to use $Q_1(A)$ instead of $\text {d}\Gamma (\cdot )$ ; the reason for this is that the relations of Proposition 3.4 below are symmetric in the Q’s.

Note that the basis-independence is a nice property of the real space setting. In general, if V is a complex Hilbert space and B is symmetric, then the definition of $Q_{2}(B)$ in (3.4) may depend on the basis. In fact, we can obtain a basis-independent formulation in the complex case, but the mapping $B\mapsto Q_{2}(B)$ is not to be defined for symmetric linear operators B, but rather symmetric anti-linear operators B to make up for the fact that in the complex case the assignment $\varphi \mapsto a(\varphi )$ is also anti-linear. This is unimportant for our application, which is why we only consider real Hilbert spaces in this section, for the sake of simplicity.

3.2 Bogolubov transformations

In this subsection, we review an explicit construction of a Bogolubov transformation $\mathcal {U}:\mathcal {F}^{+}\left(V\right)\rightarrow \mathcal {F}^{+}\left(V\right)$ that diagonalizes the quadratic Hamiltonian $H= Q_1(A)+Q_2(B)$ , namely,

(3.8) $$ \begin{align} \mathcal{U}H\mathcal{U}^{\ast}=Q_{1}(E) \end{align} $$

for a real, symmetric operator $E:V\rightarrow V$ . Such a construction is well known; see, for example, [16] for a recent review. We consider a unitary transformation $\mathcal {U}=e^{\mathcal {K}}$ where $\mathcal {K}$ is an anti-symmetric operator on $\mathcal {F}^{+}\left(V\right)$ of the following form:

(3.9) $$ \begin{align} \mathcal{K}=\frac{1}{2}\sum_{i,j=1}^{n}\langle e_{i},Ke_{j}\rangle \left(a(e_{i})a(e_{j})-a^{\ast}(e_{j})a^{\ast}(e_{i})\right)=\frac{1}{2}\sum_{i=1}^{n}\left(a(Ke_{i})a(e_{i})-a^{\ast}(e_{i})a^{\ast}(Ke_{i})\right). \end{align} $$

Here, $K:V\rightarrow V$ is a symmetric operator (called the transformation kernel) and $(e_{i})_{i=1}^{n}$ denotes any orthonormal basis of V (as with $Q_{1}(\cdot )$ and $Q_{2}(\cdot )$ this definition is independent of the basis).

In this subsection, we discuss the following:

Theorem 3.1. Let $A,B:V\rightarrow V$ be real, symmetric operators such that $A\pm B>0$ (namely, $A+B>0$ and $A-B>0$ ). Consider the Bogolubov transformation $e^{\mathcal {K}}$ where $\mathcal {K}$ is given in (3.9) with

$$\begin{align*}K=-\frac{1}{2}\log\left((A-B)^{-\frac{1}{2}}\left((A-B)^{\frac{1}{2}}(A+B)(A-B)^{\frac{1}{2}}\right)^{\frac{1}{2}}(A-B)^{-\frac{1}{2}}\right). \end{align*}$$

Then

$$ \begin{align*}e^{\mathcal{K}} (Q_1(A)+Q_2(B))e^{-\mathcal{K}}=Q_{1}(E)=2\,\mathrm{d}\Gamma(E)+\mathrm{tr}(E), \end{align*} $$

where

$$ \begin{align*}E=e^{K}(A+B)e^{K}=e^{-K}(A-B)e^{-K}. \end{align*} $$

Moreover, the diagonalizing K is uniquely determined by this.

In the following, we will prove Theorem 3.1 by using a generalization and simplification of the argument used in [23, 5]. We will first discuss the action of the Bogolubov transformation with a general kernel K and then explain where the diagonalization condition comes from.

Let us start with some basic properties of $\mathcal {K}$ .

Proposition 3.2. For any symmetric operator $K:V\rightarrow V$ , the operator $\mathcal {K}$ defined by (3.9) is an anti-symmetric operator on $\mathcal {F}^{+}\left(V\right)$ and obeys the commutators

$$ \begin{align*} \left[\mathcal{K},a(\varphi )\right] =a^{\ast}\left(K\varphi\right), \quad \left[\mathcal{K},a^{\ast}(\varphi )\right] =a\left(K\varphi\right), \quad \forall \varphi\in V. \end{align*} $$

Thus, $\left[\mathcal {K},\cdot \right]$ acts on the creation and annihilation operators by ‘swapping’ each type into the other and applying the operator K to their arguments. From this, one can now deduce that the unitary transformation $e^{\mathcal {K}}$ acts on the creation and annihilation operators according to

(3.10) $$ \begin{align} e^{\mathcal{K}}a(\varphi )e^{-\mathcal{K}} & =a\left(\cosh\left(K\right)\varphi\right)+a^{\ast}\left(\sinh\left(K\right)\varphi\right)\\ e^{\mathcal{K}}a^{\ast}(\varphi )e^{-\mathcal{K}} & =a^{\ast}\left(\cosh\left(K\right)\varphi\right)+a\left(\sinh\left(K\right)\varphi\right),\nonumber \end{align} $$

since by the Baker-Campbell-Hausdorff formula,

(3.11) $$ \begin{align} e^{\mathcal{K}}a(\varphi )e^{-\mathcal{K}} & =a(\varphi )+\frac{1}{1!}\left[\mathcal{K},a(\varphi )\right]+\frac{1}{2!}\left[\mathcal{K},\left[\mathcal{K},a(\varphi )\right]\right]+\frac{1}{3!}\left[\mathcal{K},\left[\mathcal{K},\left[\mathcal{K},a(\varphi )\right]\right]\right]+\cdots\nonumber \\ & =a(\varphi )+\frac{1}{1!}a^{\ast}\left(K\varphi\right)+\frac{1}{2!}a\left(K^{2}\varphi\right)+\frac{1}{3!}a^{\ast}\left(K^{3}\varphi\right)+\cdots\\ & =a\left(\varphi+\frac{1}{2!}K^{2}\varphi+\cdots\right)+a^{\ast}\left(\frac{1}{1!}K\varphi+\frac{1}{3!}K^{3}\varphi+\cdots\right)\nonumber \\ & =a\left(\cosh\left(K\right)\varphi\right)+a^{\ast}\left(\sinh\left(K\right)\varphi\right),\nonumber \end{align} $$

and the identity for $e^{\mathcal {K}}a^{\ast }(\varphi )e^{-\mathcal {K}}$ then follows immediately by taking the adjoint.

Now let us consider $e^{\mathcal {K}}Q_{1}(\cdot )e^{-\mathcal {K}}$ and $e^{\mathcal {K}}Q_{2}(\cdot )e^{-\mathcal {K}}$ . For this, we will first make an observation on their structure which will greatly simplify computations: namely, we note that the operators $Q_{1}(A)$ and $Q_{2}(B)$ are both of a ‘trace-form’ in the sense that we can write, say, $Q_{1}(A)=\sum _{i=1}^{n}q\left(e_{i},Ae_{i}\right)$ , where

(3.12) $$ \begin{align} q\left(x,y\right)=a^{\ast}\left(y\right)a (x )+a (x )a^{\ast}\left(y\right) \end{align} $$

defines a bilinear mapping from $V\times V$ into the space of operators on $\mathcal {F}^{+}\left(V\right)$ , similar to how the trace of an operator T is $\text {tr}(T)=\sum _{i=1}^{n}q\left(e_{i},Te_{i}\right)$ for $q\left(x,y\right)=\left\langle x,y\right\rangle $ . This abstract viewpoint is worth noting because all such expressions are both basis-independent and obey an additional property, which for the trace is just the familiar cyclicity property. Since we will encounter such ‘trace-form’ expressions repeatedly during computations throughout this paper, we state this property in full generality. In the following, we take sesquilinear to mean anti-linear in the first argument and linear in the second (we note that in the present real case a sesquilinear mapping is of course just a bilinear mapping, but stating it in this generality will prove useful later).

Lemma 3.3. Let $\left(V,\left\langle \cdot ,\cdot \right\rangle \right)$ be an n-dimensional Hilbert space and let $q:V\times V\rightarrow W$ be a sesquilinear mapping into a vector space W. Let $(e_{i})_{i=1}^{n}$ be an orthonormal basis for V. Then for any linear operators $S,T:V\rightarrow V$ , it holds that

$$\begin{align*}\sum_{i=1}^{n}q\left(Se_{i},Te_{i}\right)=\sum_{i=1}^{n}q\left(ST^{\ast}e_{i},e_{i}\right). \end{align*}$$

As a consequence, the expression $\sum _{i=1}^{n}q\left(e_{i},e_{i}\right)$ is independent of the chosen basis.

Proof. By orthonormal expansion, we find that

(3.13) $$ \begin{align} \sum_{i=1}^{n}q\left(Se_{i},Te_{i}\right) & =\sum_{i=1}^{n}q\left(Se_{i},\sum_{j=1}^{n}\left\langle e_{j},Te_{i}\right\rangle e_{j}\right) =\sum_{j=1}^{n}q\left(\sum_{i=1}^{n}\left\langle Te_{i},e_{j}\right\rangle Se_{i},e_{j}\right) \nonumber\\ & =\sum_{j=1}^{n}q\left(S\sum_{i=1}^{n}\left\langle e_{i},T^{\ast}e_{j}\right\rangle e_{i},e_{j}\right)=\sum_{i=1}^{n}q\left(ST^{\ast}e_{i},e_{i}\right). \end{align} $$

The basis independence follows from the fact that for all unitary transformation $U:V\rightarrow V$ ,

(3.14) $$ \begin{align} \sum_{i=1}^{n}q\left(Ue_{i},Ue_{i}\right)=\sum_{i=1}^{n}q\left(UU^{\ast}e_{i},e_{i}\right)=\sum_{i=1}^{n}q\left(e_{i},e_{i}\right). \end{align} $$

The lemma thus allows us to move a mapping from one argument to the other when under a sum, which will be immensely useful when simplifying expressions. As mentioned, this can indeed be seen as a generalization of the cyclicity property of the trace, since the lemma implies

(3.15) $$ \begin{align} \text{tr}\left(ST\right)=\sum_{i=1}^{n}\left\langle e_{i},STe_{i}\right\rangle =\sum_{i=1}^{n}\left\langle S^{\ast}e_{i},Te_{i}\right\rangle =\sum_{i=1}^{n}\left\langle S^{\ast}T^{\ast}e_{i},e_{i}\right\rangle =\sum_{i=1}^{n}\left\langle e_{i},TSe_{i}\right\rangle =\text{tr}\left(TS\right), \end{align} $$

but it is important to note that cyclicity is not a general property of trace-form sums; the assignments $A\mapsto Q_{1}(A)$ and $B\mapsto Q_{2}(B)$ do not obey such a property.

With the lemma, we can now easily derive the commutator of $\mathcal {K}$ with $Q_{1}(\cdot )$ and $Q_{2}(\cdot )$ :

Proposition 3.4. For any real, symmetric operators $A,B,K:V\rightarrow V$ , the operator $\mathcal {K}$ defined by equation (3.9) obeys the following commutators on $\mathcal {F}^+(V)$ :

$$ \begin{align*} \left[\mathcal{K},Q_{1}(A)\right] & =Q_{2}\left(\left\{ K,A\right\} \right)\\ \left[\mathcal{K},Q_{2}(B)\right] & =Q_{1}\left(\left\{ K,B\right\} \right). \end{align*} $$

Proof. We compute using the commutators of Proposition 3.2 that

(3.16) $$ \begin{align} \left[\mathcal{K},Q_{1}(A)\right] & =\sum_{i=1}^{n}\left(\left[\mathcal{K},a^{\ast}(Ae_{i})a(e_{i})\right]+\left[\mathcal{K},a(e_{i})a^{\ast}(Ae_{i})\right]\right)\nonumber\\ & =\sum_{i=1}^{n}(a^{\ast}(Ae_{i})\left[\mathcal{K},a(e_{i})\right] \nonumber \\ &\quad +\left[\mathcal{K},a^{\ast}(Ae_{i})\right]a(e_{i})+a(e_{i})\left[\mathcal{K},a^{\ast}(Ae_{i})\right]+\left[\mathcal{K},a(e_{i})\right]a^{\ast}(Ae_{i}))\nonumber \\ & =\sum_{i=1}^{n}\left(a^{\ast}(Ae_{i})a^{\ast}(Ke_{i})+a(KAe_{i})a(e_{i})+a(e_{i})a(KAe_{i})+a^{\ast}(Ke_{i})a^{\ast}(Ae_{i})\right). \end{align} $$

As the assignments $\varphi ,\psi \mapsto a(\varphi )a(\psi ),a^{\ast }(\varphi )a^{\ast }(\psi )$ are bilinear, we can apply Lemma 3.3 to see that

(3.17) $$ \begin{align} \left[\mathcal{K},Q_{1}(A)\right] & =\sum_{i=1}^{n}\left(a^{\ast}\left(AK^{\ast}e_{i}\right)a^{\ast}(e_{i})+a(e_{i})a\left(\left(KA\right)^{\ast}e_{i}\right)+a(e_{i})a(KAe_{i})+a^{\ast}\left(KA^{\ast}e_{i}\right)a^{\ast}(e_{i})\right)\nonumber \\ & =\sum_{i=1}^{n}\left(a^{\ast}\left(AKe_{i}\right)a^{\ast}(e_{i})+a(e_{i})a\left(AKe_{i}\right)+a(e_{i})a(KAe_{i})+a^{\ast}(KAe_{i})a^{\ast}(e_{i})\right)\nonumber \\ & =\sum_{i=1}^{n}\left(a^{\ast}\left((AK+KA)e_{i}\right)a^{\ast}(e_{i})+a(e_{i})a\left((AK+KA)e_{i}\right)\right)=Q_{2}\left(\left\{ K,A\right\} \right), \end{align} $$

where we also used that A and K are symmetric. The computation of $\left[\mathcal {K},Q_{2}(B)\right]$ is similar.

Note the similarity between this result and that of Proposition 3.2. Again we see that that $\left[\mathcal {K},\cdot \right]$ acts by ‘swapping the types and applying K to the argument’, although now the relevant types are $Q_{1}(\cdot )$ , and $Q_{2}(\cdot )$ and the application of K is taking the anticommutator.

We can now appeal to the Baker-Campbell-Hausdorff formula again to conclude that

(3.18) $$ \begin{align} &e^{\mathcal{K}}Q_{1}(A)e^{-\mathcal{K}} =Q_{1}(A)+\frac{1}{1!}\left[\mathcal{K},Q_{1}(A)\right]+\frac{1}{2!}\left[\mathcal{K},\left[\mathcal{K},Q_{1}(A)\right]\right]+\frac{1}{3!}\left[\mathcal{K},\left[\mathcal{K},\left[\mathcal{K},Q_{1}(A)\right]\right]\right]+\cdots \nonumber\\ & =Q_{1}(A)+\frac{1}{1!}Q_{2}\left(\left\{ K,A\right\} \right)+\frac{1}{2!}Q_{1}\left(\left\{ K,\left\{ K,A\right\} \right\} \right)+\frac{1}{3!}Q_{2}\left(\left\{ K,\left\{ K,\left\{ K,A\right\} \right\} \right\} \right)+\cdots \nonumber\\ & =Q_{1}\left(A+\frac{1}{2!}\left\{ K,\left\{ K,A\right\} \right\} +\cdots\right)+Q_{2}\left(\frac{1}{1!}\left\{ K,A\right\} +\frac{1}{3!}\left\{ K,\left\{ K,\left\{ K,A\right\} \right\} \right\} +\cdots\right), \end{align} $$

but to succeed, we must identify the sums of these iterated anticommutators. First, we note that we can rephrase this in a manner closer to that of equation (3.10) for $e^{\mathcal {K}}a(\varphi )e^{-\mathcal {K}}$ . One may view the anticommutator with K as a linear mapping $A\mapsto \left\{ K,A\right\} $ on the space of operators on V, $\mathcal {B}\left(V\right)$ – denote this mapping by $\mathcal {A}_{K}:\mathcal {B}\left(V\right)\rightarrow \mathcal {B}\left(V\right)$ (i.e., $\mathcal {A}_{K}(\cdot )=\left\{ K,\cdot \right\} $ ). Then we may phrase the above identity as

(3.19) $$ \begin{align} e^{\mathcal{K}}Q_{1}(A)e^{-\mathcal{K}}=Q_{1}\left(\cosh\left(\mathcal{A}_{K}\right)(A)\right)+Q_{2}\left(\sinh\left(\mathcal{A}_{K}\right)(A)\right) \end{align} $$

and likewise

(3.20) $$ \begin{align} e^{\mathcal{K}}Q_{2}(B)e^{-\mathcal{K}}=Q_{2}\left(\cosh\left(\mathcal{A}_{K}\right)(B)\right)+Q_{1}\left(\sinh\left(\mathcal{A}_{K}\right)(B)\right) \end{align} $$

so that the arguments again involve hyperbolic functions of linear operators, but now acting on $\mathcal {B}\left(V\right)$ rather than V itself. We then note the following ‘anticommutator Baker-Campbell-Hausdorff formula’:

Proposition 3.5. Let $\left(V,\left\langle \cdot ,\cdot \right\rangle \right)$ be an n-dimensional Hilbert space, let $K:V\rightarrow V$ be a self-adjoint operator and let $\mathcal {A}_{K}(\cdot )=\left\{ K,\cdot \right\} :\mathcal {B}\left(V\right)\rightarrow \mathcal {B}\left(V\right)$ denote the anticommutator with K. Then for any linear operator $T:V\rightarrow V$ ,

$$\begin{align*}e^{\mathcal{A}_{K}}(T)=\sum_{m=0}^{\infty}\frac{1}{m!}\mathcal{A}_{K}^{m}(T)=e^{K}Te^{K}. \end{align*}$$

Consequently,

$$ \begin{align*} \cosh\left(\mathcal{A}_{K}\right)(T) & =\frac{1}{2}(e^{K}Te^{K}+e^{-K}Te^{-K}),\\ \sinh(\mathcal{A}_{K})(T) & =\frac{1}{2}(e^{K}Te^{K}-e^{-K}Te^{-K}). \end{align*} $$

Proof. Let $(x_{i})_{i=1}^{n}$ be an eigenbasis for K with associated eigenvalues $\left(\lambda _{i}\right)_{i=1}^{n}$ . Denote $P_{i,j}=|x_j\rangle \langle x_i|$ , namely, $P_{i,j}x=\left\langle x_{i},x\right\rangle x_{j}$ for all $x\in V$ . It is well known that for any orthonormal basis $(x_{i})_{i=1}^{n}$ of V, the collection $\left(P_{i,j}\right)_{i,j=1}^{n}$ form an orthonormal basis for $\left(\mathcal {B}\left(V\right),\left\langle \cdot ,\cdot \right\rangle _{\text {HS}}\right)$ . Moreover, for any $x\in V$ and $1\leq i,j\leq n$ , by self-adjointness of K,

(3.21) $$ \begin{align} \mathcal{A}_{K}\left(P_{i,j}\right)x & =\left\{ K,P_{i,j}\right\} x=\left\langle x_{i},x\right\rangle Kx_{j}+\left\langle x_{i},Kx\right\rangle x_{j}=\left\langle x_{i},x\right\rangle \lambda_{j}x_{j}+\left\langle \lambda_{i}x_{i},x\right\rangle x_{j}\\ & =\left(\lambda_{i}+\lambda_{j}\right)\left\langle x_{i},x\right\rangle x_{j}=\left(\lambda_{i}+\lambda_{j}\right)P_{i,j}x.\nonumber \end{align} $$

Thus, $\{P_{i,j}\}_{i,j=1}^n$ an eigenbasis for $\mathcal {A}_{K}$ with associated eigenvalues $\left(\lambda _{i}+\lambda _{j}\right)_{i,j=1}^{n}$ .

Hence, it suffices to verify the identity $e^{\mathcal {A}_{K}}(T)=e^{K}Te^{K}$ with the eigenbasis $\left(P_{i,j}\right)_{i,j=1}^{n}$ :

(3.22) $$ \begin{align} e^{\mathcal{A}_{K}}\left(P_{i,j}\right) x & =e^{\lambda_{i}+\lambda_{j}}P_{i,j}=e^{\lambda_{i}+\lambda_{j}}\left\langle x_{i},x\right\rangle x_{j}=\left\langle e^{\lambda_{i}}x_{i},x\right\rangle e^{\lambda_{j}}x_{j}=\left\langle e^{K}x_{i},x\right\rangle e^{K}x_{j}\\ & =\left\langle x_{i},e^{K}x\right\rangle e^{K}x_{j}=e^{K}P_{i,j}e^{K}x.\nonumber \end{align} $$

The statements regarding $\cosh \left(\mathcal {A}_{K}\right)$ and $\sinh \left(\mathcal {A}_{K}\right)$ follow from the identities

(3.23) $$ \begin{align} \cosh (x )=\frac{1}{2}\left(e^{x}+e^{-x}\right), \quad \sinh (x )=\frac{1}{2}\left(e^{x}-e^{-x}\right), \quad \text{and }\left(-\mathcal{A}_{K}\right)=\mathcal{A}_{-K}. \end{align} $$

By these formulas, we thus deduce the quadratic operator analogue of equation (3.10):

(3.24) $$ \begin{align} e^{\mathcal{K}}Q_{1}(A)e^{-\mathcal{K}} & =\frac{1}{2}Q_{1}(e^{K}Ae^{K}+e^{-K}Ae^{-K})+\frac{1}{2}Q_{2}(e^{K}Ae^{K}-e^{-K}Ae^{-K}) \nonumber\\ e^{\mathcal{K}}Q_{2}(B)e^{-\mathcal{K}} & =\frac{1}{2}Q_{1}(e^{K}Be^{K}-e^{-K}Be^{-K})+\frac{1}{2}Q_{2}(e^{K}Be^{K}+e^{-K}Be^{-K}). \end{align} $$

Diagonalization condition

We can now finally describe how to diagonalize a quadratic Hamiltonian using a Bogolubov transformation of the form $e^{\mathcal {K}}$ . By the transformation identities above, we find that under $e^{\mathcal {K}}$ , the quadratic Hamiltonian $H=Q_{1}(A)+Q_{2}(B)$ transforms as

(3.25) $$ \begin{align} e^{\mathcal{K}}He^{-\mathcal{K}} &=\frac{1}{2}Q_{1}(e^{K}Ae^{K}+e^{-K}Ae^{-K})+\frac{1}{2}Q_{2}(e^{K}Ae^{K}-e^{-K}Ae^{-K})\nonumber \\ &\quad +\frac{1}{2}Q_{1}(e^{K}Be^{K}-e^{-K}Be^{-K})+\frac{1}{2}Q_{2}(e^{K}Be^{K}+e^{-K}Be^{-K})\nonumber\\ & =\frac{1}{2}Q_{1}(e^{K}(A+B)e^{K}+e^{-K}(A-B)e^{-K})+\frac{1}{2}Q_{2}(e^{K}(A+B)e^{K}-e^{-K}(A-B)e^{-K}). \end{align} $$

Therefore, the diagonalization condition on K is

(3.26) $$ \begin{align} e^{K}(A+B)e^{K}=e^{-K}(A-B)e^{-K}. \end{align} $$

If we can find such a K, then

(3.27) $$ \begin{align} e^{\mathcal{K}}He^{-\mathcal{K}}=Q_{1}(E)=2\,\text{d}\Gamma(E)+\text{tr}(E), \end{align} $$

where

(3.28) $$ \begin{align} E=e^{K}(A+B)e^{K}=e^{-K}(A-B)e^{-K}. \end{align} $$

There remains the question of existence and uniqueness of such a K:

Conclusion of the proof of Theorem 3.1 .

Write $A_{\pm }=A\pm B>0$ for brevity. Then we may write the diagonalization condition as

(3.29) $$ \begin{align} e^{-2K}A_{-}e^{-2K}=A_{+}. \end{align} $$

Multiplying by $A_{-}^{\frac {1}{2}}$ on both sides yields

(3.30) $$ \begin{align} \big(A_{-}^{\frac{1}{2}}e^{-2K}A_{-}^{\frac{1}{2}}\big)^{2}=A_{-}^{\frac{1}{2}}e^{-2K}A_{-}e^{-2K}A_{-}^{\frac{1}{2}}=A_{-}^{\frac{1}{2}}A_{+}A_{-}^{\frac{1}{2}}, \end{align} $$

which is equivalent to

(3.31) $$ \begin{align} A_{-}^{\frac{1}{2}}e^{-2K}A_{-}^{\frac{1}{2}}=\big(A_{-}^{\frac{1}{2}}A_{+}A_{-}^{\frac{1}{2}}\big)^{\frac{1}{2}}, \, \text{ namely } e^{-2K}=A_{-}^{-\frac{1}{2}}\big(A_{-}^{\frac{1}{2}}A_{+}A_{-}^{\frac{1}{2}}\big)^{\frac{1}{2}}A_{-}^{-\frac{1}{2}}. \end{align} $$

This implies the existence and uniqueness of the diagonalizing K as the operator exponential is a bijection between the real, symmetric operators and the real, symmetric, positive-definite operators.

4 The quasi-bosonic quadratic Hamiltonian

Now we turn to the quasi-bosonic setting. We start by casting the bosonizable terms $ H_{\operatorname {\mathrm {kin}}}^{\prime } + \sum _{k\in S_C} H^k_{\mathrm {int}}$ , which we encountered in Section 2.1, into a form which closely mirrors the form of the bosonic quadratic Hamiltonians that we considered in the preceding section.

4.1 Quadratic Hamiltonian

Let us define the pair excitation operators

(4.1) $$ \begin{align} b_{k,p}=c_{p-k}^{\ast}c_{p},\quad b_{k,p}^{\ast}=c_{p}^{\ast}c_{p-k},\quad k\in\mathbb{Z}_{\ast}^{3},\quad p\in L_{k}. \end{align} $$

We remark that in contrast to the bosonic case, the fermionic creation and annihilation operators are bounded (in fact, $\Vert c_{p,\sigma }\Vert _{\text {Op}}=\Vert c_{p,\sigma }^{\ast }\Vert _{\text {Op}}=1$ ), and therefore so are the operators $b_{k,p}^{\ast }$ , $b_{k,p}$ .

Then $H_{\text {int}}^{k}$ in (1.34) is exactly given by

(4.2) $$ \begin{align} H_{\text{int}}^{k} & =\sum_{p,q\in L_{k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left(b_{k,p}^{\ast}b_{k,q}+b_{k,q}b_{k,p}^{\ast}\right)+\sum_{p,q\in L_{-k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left(b_{-k,p}^{\ast}b_{-k,q}+b_{-k,q}b_{-k,p}^{\ast}\right)\nonumber \\ &\quad +\sum_{p\in L_{k}}\sum_{q\in L_{-k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left(b_{k,p}^{\ast}b_{-k,q}^{\ast}+b_{-k,q}b_{k,p}\right) \nonumber \\ &\quad +\sum_{p\in L_{-k}}\sum_{q\in L_{k}}\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left(b_{-k,p}^{\ast}b_{k,q}^{\ast}+b_{k,q}b_{-k,p}\right). \end{align} $$

Thus, the natural one-body Hilbert space associated to $H_{\text {int}}^{k}$ is $\ell ^2 (L_k \cup L_{-k})$ . To free us from having to explicitly write sums over $L_{k}$ and $L_{-k}$ separately, we introduce some more notation. First, we will denote this union of lunes by

(4.3) $$ \begin{align} L_{k}^{\pm}=L_{k}\cup L_{-k}, \quad \ell^{2}\left(L_{k}^{\pm}\right)=\ell^{2}\left(L_{k}\cup L_{-k}\right)=\ell^{2}(L_{k})\oplus\ell^{2}(L_{-k}), \quad k\in\mathbb{Z}_{+}^{3}.\end{align} $$

Here, we used the fact that $L_{k}\cap L_{-k}=\emptyset $ for any $k\in \mathbb {Z}_{+}^{3}$ , since if $p\in L_{k}\cap L_{-k}$ , then

$$ \begin{align*}2|p|^2 \geq |p-k|^2 + |p+k|^2 = 2|p|^2 + 2|k|^2> 2|p|^2, \end{align*} $$

which is a contradiction. It is also convenient to introduce the ‘bar-notation’

(4.4) $$ \begin{align} \overline{k,p}=\begin{cases} k,p & p\in L_{k}\\ -k,p & p\in L_{-k} \end{cases},\quad\overline{p-k}=\begin{cases} p-k & p\in L_{k}\\ p+k & p\in L_{-k} \end{cases} \end{align} $$

to automatically encode the appropriate sign of k depending on $p\in L_{k}^{\pm }=L_{k}\cup L_{-k}$ (this will allow us to avoid expanding all our terms on a case-by-case basis when this is irrelevant).

In analogy with the definitions (3.3) and (3.4) we now define, for any $k\in \mathbb {Z}_{+}^{3}$ and symmetric operators $A,B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , the quadratic operators $Q_{1}^{k}(A),Q_{2}^{k}(B):\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ by

(4.5) $$ \begin{align} Q_{1}^{k}(A) & =\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{p},Ae_{q}\right\rangle \left(b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}+b_{\overline{k,q}}b_{\overline{k,p}}^{\ast}\right),\nonumber\\ Q_{2}^{k}(B) & =\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{p},Be_{q}\right\rangle \left(b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right). \end{align} $$

In order to cast $H_{\text {int}}^{k}$ as given by equation (4.2) into this form, we must identify the relevant operators A and B. Define the (un-normalized) rank-one projection $P_{v_k}: \ell ^2(L_k)\to \ell ^2(L_k)$ by

(4.6) $$ \begin{align} P_{v_k} =|v_k\rangle \langle v_k|, \quad v_{k}=\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sum_{p\in L_{k}}e_{p} \in \ell^2(L_k), \end{align} $$

where $(e_{p})_{p\in L_{k}}$ denotes the standard orthonormal basis of $\ell ^{2}(L_{k})$ . Put differently, the matrix elements of $P_{v_k}$ are $\left\langle e_{p},P_{v_{k}}e_{q}\right\rangle =\frac {1}{2\left(2\pi \right)^{3}} \hat {V}_{k}k_{F}^{-1}$ for all $p,q\in L_{k}$ . Next, we define the operators

(4.7) $$ \begin{align} A_{k}^{\oplus},B_{k}^{\oplus}:\ell^{2}\left(L_{k}^{\pm}\right)\rightarrow\ell^{2}\left(L_{k}^{\pm}\right), \quad A_{k}^{\oplus}=\left(\begin{array}{cc} P_{v_{k}} & 0\\ 0 & P_{v_{k}} \end{array}\right),\quad B_{k}^{\oplus}=\left(\begin{array}{cc} 0 & P_{v_{k}}\\ P_{v_{k}} & 0 \end{array}\right) \end{align} $$

with respect to the decomposition $\ell ^{2}(L_{k}^{\pm })=\ell ^{2}(L_{k})\oplus \ell ^{2}(L_{-k})$ and the identification $\ell ^{2}(L_{k})\cong \ell ^{2}(L_{-k})$ (under $e_{p}\mapsto e_{-p}$ ).

Thus, the operator $H_{\text {int}}^{k}$ is concisely expressed as

(4.8) $$ \begin{align} H_{\text{int}}^{k}=Q_{k}^{1}\left(A_{k}^{\oplus}\right)+Q_{k}^{2}\left(B_{k}^{\oplus}\right). \end{align} $$

It remains to consider the kinetic operator. The equality (1.74) bids us to think of $H_{\operatorname {\mathrm {kin}}}^{\prime }$ as it were

(4.9) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime} &\sim \sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\left(|p|^{2}-|p-k|^{2}\right)b_{k,p}^{\ast}b_{k,p} \nonumber\\ &= \sum_{k\in\mathbb{Z}_{+}^{3}} \left(\sum_{p\in L_{k}}\left(|p|^{2}-|p-k|^{2}\right)b_{k,p}^{\ast}b_{k,p}+\sum_{p\in L_{-k}}\left(|p|^{2}-\left|p+k\right|^{2}\right)b_{-k,p}^{\ast}b_{-k,p}\right) \end{align} $$

in an appropriate sense. To put this in the same framework as $H_{\mathrm {int}}^k$ , let us introduce (for every $k\in \mathbb {Z}_{+}^{3}$ ) the operator $h_{k}:\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ by

(4.10) $$ \begin{align} h_{k}e_{p} =\lambda_{k,p}e_{p},\quad\;\;\lambda_{k,p}=\frac{1}{2}(|p|^{2}-|p-k|^{2}). \end{align} $$

Using again the identification $\ell ^{2}(L_{k})\cong \ell ^{2}(L_{-k})$ (under $e_{p}\mapsto e_{-p}$ ), we define the operators $h_{k}^{\oplus }:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ by

(4.11) $$ \begin{align} h_{k}^{\oplus}=\left(\begin{array}{cc} h_{{k}} & 0\\ 0 & h_{{k}} \end{array}\right). \end{align} $$

Then we can rewrite (4.9) as

(4.12) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime} \sim \sum_{k\in\mathbb{Z}_{+}^{3}} \Big( Q_1^k (h_k^{\oplus}) - 2\,\mathrm{tr} (h_k)\Big). \end{align} $$

Recall that $S_{C}=\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ for an exponent $1\ge \gamma>0$ which is to be optimized over at the end. As far as the lower bound is concerned, we may replace $\sum _{k\in \mathbb {Z}_+}$ by $\sum _{k\in S_C}$ (the upper bound is easier and will be explained separately). In summary, we arrive at the following quasi-bosonic expression for the bosonizable terms:

(4.13) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime} + \sum_{k\in S_C} H_{\mathrm{int}}^k \sim \sum_{k\in S_C} \Big( Q_1^k (h_k^{\oplus} + A_k^{\oplus}) + Q_2^k (B_k^{\oplus}) - 2\,\mathrm{tr}(h_k)\Big). \end{align} $$

Note that unlike the bosonic case, the operators on the right side of (4.13) are bounded.

4.2 Generalized pair operators

For every $k\in \mathbb {Z}_{+}^{3}$ and $\varphi \in \ell ^2(L_k^{\pm })$ , we define the operators

(4.14) $$ \begin{align} b_{k}(\varphi )=\sum_{p\in L_{k}^{\pm}}\left\langle \varphi,e_{p}\right\rangle b_{\overline{k,p}},\quad b_{k}^{\ast}(\varphi )=\sum_{p\in L_{k}^{\pm}}\left\langle e_{p},\varphi\right\rangle b_{\overline{k,p}}^{\ast}. \end{align} $$

They obey the quasi-bosonic commutation relations (for $k,l\in \mathbb {Z}_{+}^{3}$ and $\varphi \in \ell ^{2}(L_{k}^{\pm })$ , $\psi \in \ell ^{2}\left(L_{l}^{\pm }\right)$ )

(4.15) $$ \begin{align} \left[b_{k}(\varphi ),b_{l}\left(\psi\right)\right] & =\left[b_{k}^{\ast}(\varphi ),b_{l}^{\ast}\left(\psi\right)\right]=0, \nonumber\\ \left[b_{k}(\varphi ),b_{l}^{\ast}\left(\psi\right)\right] & =\delta_{k,l}\left\langle \varphi,\psi\right\rangle +\varepsilon_{k,l}\left(\varphi;\psi\right), \end{align} $$

where the correction term is

(4.16) $$ \begin{align} \varepsilon_{k,l}\left(\varphi;\psi\right) & =\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\psi\right\rangle \varepsilon\left(\overline{k,p};\overline{l,q}\right), \nonumber\\ \varepsilon\left(\overline{k,p};\overline{l,q}\right) & =-\left(\delta_{p,q}c_{\overline{q-l}}c_{\overline{p-k}}^{\ast}+\delta_{\overline{p-k},\overline{q-l}}c_{q}^{\ast}c_{p}\right). \end{align} $$

We simply have $b_{k}(e_{p})=b_{\overline {k,p}}$ and the quadratic operators in (4.5) can be expressed as

(4.17) $$ \begin{align} Q_{1}^{k}(A) & =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})b_{k}(e_{p})+b_{k}(e_{p})b_{k}^{\ast}(Ae_{p})\right) \nonumber\\ Q_{2}^{k}(B) & =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}\left(Be_{p}\right)b_{k}^{\ast}(e_{p})+b_{k}(e_{p})b_{k}\left(Be_{p}\right)\right) \end{align} $$

in analogy with equation (3.5). In order to justify the quasi-bosonic interpretation, we need rigorous estimates for the correction term in (4.16). Let us start with the following:

Proposition 4.1. For all $k\in \mathbb {Z}_{+}^{3}$ and $\varphi \in \ell ^{2}(L_{k}^{\pm })$ , it holds that $\varepsilon _{k,k}\left(\varphi ,\varphi \right)\leq 0$ , namely,

$$ \begin{align*}b_{k}(\varphi )b_{k}^{\ast}(\varphi ) \le b_{k}^{\ast}(\varphi )b_{k}(\varphi )+\left\Vert \varphi\right\Vert ^{2}. \end{align*} $$

Note that the observation of the error term $\varepsilon _{k,k}$ being non-positive also appeared in [5, Proof of Lemma 4.2] in the context of different bosonic operators.

Proof. We expand the term

(4.18) $$ \begin{align} \varepsilon_{k,k}\left(\varphi;\varphi\right) & =\sum_{p,q\in L_{k}^{\pm}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle \varepsilon\left(\overline{k,p};\overline{k,q}\right) \nonumber\\ &=-\sum_{p,q\in L_{k}^{\pm}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle \left(\delta_{p,q}c_{\overline{q-k}}c_{\overline{p-k}}^{\ast}+\delta_{\overline{p-k},\overline{q-k}}c_{q}^{\ast}c_{p}\right)\nonumber \\ & =-\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi\right\rangle \right|^{2}c_{\overline{p-k}}c_{\overline{p-k}}^{\ast}-\sum_{p,q\in L_{k}^{\pm}}\delta_{\overline{p-k},\overline{q-k}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle c_{q}^{\ast}c_{p} \\ &\le - \sum_{p,q\in L_{k}^{\pm}}\delta_{\overline{p-k},\overline{q-k}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle c_{q}^{\ast}c_{p}.\nonumber \end{align} $$

We treat the terms of the last sum on a case-by-case basis according to which of $L_{k}$ and $L_{-k}$ , p and q lie in: if p and q lie in the same lune, then $\delta _{\overline {p-k},\overline {q-k}}=\delta _{p\mp k,q\mp k}=\delta _{p,q}$ and so

(4.19) $$ \begin{align} A =\left(\sum_{p,q\in L_{k}}+\sum_{p,q\in L_{-k}}\right)\delta_{\overline{p-k},\overline{q-k}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle c_{q}^{\ast}c_{p}=\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi\right\rangle \right|^{2}c_{p}^{\ast}c_{p} \ge 0. \end{align} $$

However, by the Cauchy–Schwarz inequality,

(4.20) $$ \begin{align} &\pm \left( \sum_{p\in L_{k}}\sum_{q\in L_{-k}} + \sum_{p\in L_{-k}}\sum_{q\in L_{k}} \right) \delta_{\overline{p-k},\overline{q-k}}\left\langle \varphi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle c_{q}^{\ast}c_{p}\nonumber\\ &\le \left( \sum_{p\in L_{k}}\sum_{q\in L_{-k}} + \sum_{p\in L_{-k}}\sum_{q\in L_{k}} \right) \frac{1}{2}\Big( \delta_{\overline{p-k},\overline{q-k}} |\left\langle \varphi,e_{p}\right\rangle|^2 c_p^* c_p + |\left\langle e_{q},\varphi\right\rangle|^2 c_{q}^{\ast}c_{q} \Big) \\ &\le \sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi\right\rangle \right|^{2}c_{p}^{\ast}c_{p} = A. \nonumber \end{align} $$

We thus conclude that $\varepsilon _{k,k}\left(\varphi ,\varphi \right)\leq 0$ as claimed.

Next, we have the following:

Proposition 4.2. For all $k\in \mathbb {Z}_{+}^{3}$ , $\varphi \in \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\left\Vert b_{k}(\varphi )\Psi\right\Vert \leq\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle },\quad\left\Vert b_{k}^{\ast}(\varphi )\Psi\right\Vert \leq\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle }. \end{align*}$$

The bounds here are similar to [5, Lemma 4.2]. Recall that in our quasi-bosonic setting the excitation number operator

(4.21) $$ \begin{align} \mathcal{N}_{E}=\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}=\sum_{p\in B_{F}}c_{p}c_{p}^{\ast} \end{align} $$

plays the role that the usual number operator $\mathcal {N}$ does in the exact bosonic case. Thus, Proposition 4.2 is the analogue of the well-known bosonic estimate

(4.22) $$ \begin{align} \left\Vert a(\varphi )\Psi\right\Vert \leq\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\mathcal{N}\Psi\right\rangle },\quad\left\Vert a^{\ast}(\varphi )\Psi\right\Vert \leq\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\left(1+\mathcal{N}\right)\Psi\right\rangle }. \end{align} $$

Proof. By the Cauchy-Schwarz inequality,

(4.23) $$ \begin{align} \left\Vert b_{k}(\varphi )\Psi\right\Vert & = \left\Vert \sum_{p\in L_{k}^{\pm}}\left\langle \varphi,e_{p}\right\rangle b_{\overline{k,p}}\Psi\right\Vert \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\left|\left\langle \varphi,e_{p}\right\rangle \right|^{2}}\sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}}\nonumber \\ & \leq\left\Vert \varphi\right\Vert \sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert c_{p}\Psi\right\Vert ^{2}}\leq\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle }. \end{align} $$

The second bound follows from the first and Proposition 4.1.

We remark that the above estimate is also valid for $\Psi \in \mathcal {H}_{M}$ when $M\neq N$ , provided $\mathcal {N}_{E}$ is understood as $\sum _{p\in B_{F}^{c}}c_{p}^{\ast }c_{p}$ acting on $\mathcal {H}_{M}$ (in (4.23) we used $L_k^{\pm } \subset B_F^c$ ). One must be precise here as the identity $\mathcal {N}_{E}=\sum _{p\in B_{F}}c_{p}c_{p}^{\ast }$ does not hold on $\mathcal {H}_{M}$ . In fact, the estimate also holds if $\mathcal {N}_{E}$ is understood as $\sum _{p\in B_{F}}c_{p}c_{p}^{\ast }$ , up to an additional factor of $\sqrt {2}$ due to the necessary overcounting of the holes, ⁷ namely, from $\left\Vert b_{\overline {k,p}}\Psi \right\Vert =\left\Vert c_{\overline {p-k}}^{\ast }c_{p}\Psi \right\Vert \leq \left\Vert c_{\overline {p-k}}^{\ast }\Psi \right\Vert $ with $\overline {p-k} \in B_F$ , we get

(4.24) $$ \begin{align} \left\Vert b_{k}(\varphi )\Psi\right\Vert \leq\left\Vert \varphi\right\Vert \sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert c_{\overline{p-k}}^{\ast}\Psi\right\Vert ^{2}}\leq\sqrt{2}\left\Vert \varphi\right\Vert \sqrt{\left\langle \Psi,\left(\sum_{p\in B_{F}}c_{p}c_{p}^{\ast}\right)\Psi\right\rangle }. \end{align} $$

This is a point that we must consider, since below we will also encounter expressions such as $\left\Vert b_{k}(\varphi )c_{p}\Psi \right\Vert $ for $\Psi \in \mathcal {H}_{N}$ (so that $c_{p}\Psi \in \mathcal {H}_{N-1}$ ). For this, we denote by $\mathcal {N}_{E}^{\left(-1\right)}:\mathcal {H}_{N-1}\rightarrow \mathcal {H}_{N-1}$ and $\mathcal {N}_{E}^{\left(+1\right)}:\mathcal {H}_{N+1}\rightarrow \mathcal {H}_{N+1}$ the operators

(4.25) $$ \begin{align} \mathcal{N}_{E}^{\left(-1\right)}=\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p},\quad\mathcal{N}_{E}^{\left(+1\right)}=\sum_{p\in B_{F}}c_{p}c_{p}^{\ast}. \end{align} $$

This choice is motivated by the following identities:

Lemma 4.3. For all $p\in B_{F}^{c}$ and $q\in B_{F}$ , it holds that

$$ \begin{align*} \mathcal{N}_{E}c_{p}^{\ast} &=c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}+c_{p}^{\ast}, \quad c_p\mathcal{N}_{E}^{(-1)}c_{p}^{\ast} \le \mathcal{N}_E,\\ \mathcal{N}_{E}c_{q} &=c_{q}\mathcal{N}_{E}^{\left(+1\right)}+c_{q}, \quad c_{q}^{\ast}\mathcal{N}_{E}^{\left(+1\right)} c_{q} \le \mathcal{N}_E. \end{align*} $$

Consequently,

$$\begin{align*}\sum_{p\in B_{F}^{c}}c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}c_{p}=\mathcal{N}_{E}^{2}-\mathcal{N}_{E}=\sum_{p\in B_{F}}c_{p}\mathcal{N}_{E}^{\left(+1\right)}c_{p}^{\ast}. \end{align*}$$

Proof. This follows directly by the CAR, as for all $p\in B_{F}^{c}$ ,

(4.26) $$ \begin{align} \mathcal{N}_{E}c_{p}^{\ast}=\sum_{q\in B_{F}^{c}}c_{q}^{\ast}c_{q}c_{p}^{\ast}=\sum_{q\in B_{F}^{c}}c_{p}^{\ast}c_{q}^{\ast}c_{q}+\sum_{q\in B_{F}^{c}}\left(c_{q}^{\ast}\left\{ c_{q},c_{p}^{\ast}\right\} -\left\{ c_{q}^{\ast},c_{p}^{\ast}\right\} c_{q}\right)=c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}+c_{p}^{\ast}. \end{align} $$

Consequently, using $\|c_p\|_{\mathrm {Op}}=1$ and $[\mathcal {N}_{E},c_{p}^{\ast }c_p]=0$ , we have

(4.27) $$ \begin{align} \mathcal{N}_{E} \ge \mathcal{N}_{E}c_{p}^{\ast}c_p = c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}c_p + c_{p}^{\ast} c_p \ge c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}c_p. \end{align} $$

Likewise, for all $q\in B_{F}$ ,

(4.28) $$ \begin{align} \mathcal{N}_{E}c_{q}=\sum_{p\in B_{F}}c_{p}c_{p}^{\ast}c_{q}=\sum_{p\in B_{F}}c_{q}c_{p}c_{p}^{\ast}+\sum_{p\in B_{F}}\left(c_{p}\left\{ c_{p}^{\ast},c_{q}\right\} -\left\{ c_{p},c_{q}\right\} c_{p}^{\ast}\right)=c_{q}\mathcal{N}_{E}^{\left(+1\right)}+c_{q}, \end{align} $$

and hence, $\mathcal {N}_{E} \ge c_{q}^{\ast }\mathcal {N}_{E}^{\left(+1\right)} c_{q}$ . Moreover,

(4.29) $$ \begin{align} \sum_{p\in B_{F}^{c}}c_{p}^{\ast}\mathcal{N}_{E}^{\left(-1\right)}c_{p} &=\sum_{p\in B_{F}^{c}}\left(\mathcal{N}_{E}c_{p}^{\ast}-c_{p}^{\ast}\right)c_{p}=\mathcal{N}_{E}^{2}-\mathcal{N}_{E}=\sum_{p\in B_{F}}\left(\mathcal{N}_{E}c_{p}-c_{p}\right)c_{p}^{\ast}\nonumber \\ &=\sum_{p\in B_{F}}c_{p}\mathcal{N}_{E}^{\left(+1\right)}c_{p}^{\ast}. \end{align} $$

In some cases, it is important to refine error estimates by using the kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ rather than $\mathcal {N}_{E}$ . We can implement the kinetic estimate of Proposition 2.2 in the generalized setting:

Proposition 4.4. For all $k\in \mathbb {Z}_{+}^{3}$ , $\varphi \in \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ , it holds that

$$\begin{align*}\left\Vert b_{k}(\varphi )\Psi\right\Vert \leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\varphi\right\Vert \sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle },\quad\left\Vert b_{k}^{\ast}(\varphi )\Psi\right\Vert \leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\varphi\right\Vert \sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }+\left\Vert \varphi\right\Vert \left\Vert \Psi\right\Vert. \end{align*}$$

Proof. We start by applying the Cauchy-Schwarz inequality

(4.30) $$ \begin{align} \left\Vert b_{k}(\varphi )\Psi\right\Vert =\left\Vert \sum_{p\in L_{k}^{\pm}}\left\langle \varphi,e_{p}\right\rangle b_{\overline{k,p}}\Psi\right\Vert \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}^{-1}\left|\left\langle \varphi,e_{p}\right\rangle \right|^{2}}\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}}. \end{align} $$

As the vectors $(e_{p})_{p\in L_{k}^{\pm }}$ obey $h_{k}^{\oplus }e_{p}=\lambda _{\overline {k,p}}e_{p}$ , we recognize the first sum on the right-hand side as

(4.31) $$ \begin{align} \sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}^{-1}\left|\left\langle \varphi,e_{p}\right\rangle \right|^{2}=\left\langle \varphi,\left(h_{k}^{\oplus}\right)^{-1}\varphi\right\rangle =\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\varphi\right\Vert ^{2}. \end{align} $$

For the second sum, we have by equation (2.4) that

(4.32) $$ \begin{align} \sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}=\sum_{p\in L_{k}}\lambda_{k,p}\left\Vert c_{p-k}^{\ast}c_{p}\Psi\right\Vert ^{2}+\sum_{p\in L_{-k}}\lambda_{-k,p}\left\Vert c_{p+k}^{\ast}c_{p}\Psi\right\Vert ^{2}\leq\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle, \end{align} $$

which implies the first claim. The second bound follows from the first and Proposition 4.1:

(4.33) $$ \begin{align} \left\Vert b_{k}^{\ast}(\varphi )\Psi\right\Vert \leq\sqrt{\left\langle \Psi,\left(\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\varphi\right\Vert ^{2}H_{\operatorname{\mathrm{kin}}}^{\prime}+\left\Vert \varphi\right\Vert ^{2}\right)\Psi\right\rangle }\leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\varphi\right\Vert \sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }+\left\Vert \varphi\right\Vert \left\Vert \Psi\right\Vert. \end{align} $$

4.3 Preliminary estimates for quadratic operators

In this subsection, we provide some basic bounds on the quadratic operators $Q_{1}^{k}(A)$ and $Q_{1}^{k}(B)$ defined in (4.5) for any $k\in \mathbb {Z}_{+}^{3}$ . First, for $Q_{1}^{k}(A)$ , we can normal order as follows:

(4.34) $$ \begin{align} Q_{1}^{k}(A) & =\sum_{p\in L_{k}^{\pm}}\left(2\,b_{k}^{\ast}(Ae_{p})b_{k}(e_{p})+\left[b_{k}(e_{p}),b_{k}^{\ast}(Ae_{p})\right]\right)\nonumber \\ & =2\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}(Ae_{p})b_{k}(e_{p})+\sum_{p\in L_{k}^{\pm}}\left\langle e_{p},Ae_{p}\right\rangle +\sum_{p\in L_{k}^{\pm}}\varepsilon_{k,k}\left(e_{p};Ae_{p}\right)\\ & =2\,\tilde{Q}_{1}^{k}(A)+\text{tr}(A)+\varepsilon_{k}(A)\nonumber, \end{align} $$

where for brevity, we have defined the notation

(4.35) $$ \begin{align} \tilde{Q}_{1}^{k}(A)=\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}(Ae_{p})b_{k}(e_{p}),\quad\varepsilon_{k}(A)=\sum_{p\in L_{k}^{\pm}}\varepsilon_{k,k}\left(e_{p};Ae_{p}\right). \end{align} $$

The term $\tilde {Q}_{1}^{k}(A)$ plays the same role of $\text {d}\Gamma (A)$ in the exact bosonic case, whereas $\varepsilon _{k}(A)$ is a correction term in the quasi-bosonic case.

Proposition 4.5. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$ \begin{align*} \left|\left\langle \Psi,\tilde{Q}_{1}^{k}(A)\Psi\right\rangle \right| &\leq \left\Vert A\right\Vert _{\operatorname{\mathrm{Op}}}\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle ,\\ \left|\left\langle \Psi,\varepsilon_{k}(A)\Psi\right\rangle \right| &\leq 3\left\Vert A\right\Vert _{\operatorname{\mathrm{Op}}}\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align*} $$

If furthermore, $A\geq 0$ , then also $\tilde {Q}_{1}^{k}(A)\geq 0$ .

Proof. Let $(x_{i})_i$ be an eigenbasis for A with eigenvalues $\left(\lambda _{i}\right)_i$ . Noting that the mapping $x,y\mapsto b_{k}^{\ast }\left(Ax\right)b_{k}\left(y\right)$ is bilinear, we may invoke Lemma 3.3 (the part of basis independence) to write

(4.36) $$ \begin{align} \tilde{Q}_{1}^{k}(A)=\sum_{i}b_{k}^{\ast}\left(Ax_{i}\right)b_{k}(x_{i})=\sum_{i}\lambda_{i}b_{k}^{\ast}(x_{i})b_{k}(x_{i}). \end{align} $$

Clearly, if $A\ge 0$ , then all $\lambda _i\ge 0$ , and hence, $\tilde {Q}_{1}^{k}(A)\ge 0$ . In general, we always have $|\lambda _i | \le \left\Vert A\right\Vert _{\operatorname {\mathrm {Op}}}$ for all i. Hence, using Lemma 3.3 again and $b_{\overline {k,p}}^* b_{\overline {k,p}} \le c_p^* c_p$ , we have

(4.37) $$ \begin{align} \pm \tilde{Q}_{1}^{k}(A) \le \left\Vert A\right\Vert _{\text{Op}} \sum_{i}b_{k}^{\ast}(x_{i})b_{k}(x_{i})= \left\Vert A\right\Vert _{\text{Op}} \sum_{p\in L_{k}^{\pm}}b_{\overline{k,p}}^{\ast}b_{\overline{k,p}} \le \left\Vert A\right\Vert _{\text{Op}} \sum_{p\in L_{k}^{\pm}} c_p^* c_p \le \left\Vert A\right\Vert _{\text{Op}} \mathcal{N}_{E}. \end{align} $$

Similarly,

(4.38) $$ \begin{align} \pm \varepsilon_{k}(A)&= \pm \sum_{i}\varepsilon_{k,k}\left(x_{i};Ax_{i}\right)= \pm \sum_{i}\lambda_{i}\varepsilon_{k,k}\left(x_{i};x_{i}\right) \nonumber\\ &\le - \|A\|_{\mathrm{Op}}\sum_{i} \varepsilon_{k,k}\left(x_{i};x_{i}\right) = - \|A\|_{\mathrm{Op}} \sum_{p\in L_{k}^{\pm}}\varepsilon_{k,k}\left(e_{p};e_{p}\right), \end{align} $$

where in the first inequality we used the fact that $\varepsilon _{k,k}\left(x_{i};x_{i}\right)\le 0$ as shown in the proof of Proposition 4.1. Using $\varepsilon _{k,k}(e_{p};e_{p})=\varepsilon (\overline {k,p};\overline {l,q})$ and the definition (4.16), we get

(4.39) $$ \begin{align} - \sum_{p\in L_{k}^{\pm}}\varepsilon\left(\overline{k,p};\overline{k,p}\right) &=\sum_{p\in L_{k}^{\pm}}\left(c_{\overline{p-k}}c_{\overline{p-k}}^{\ast}+c_{p}^{\ast}c_{p}\right) \leq 2 \sum_{p\in B_{F}} c_{p}c_{p}^{\ast} +\sum_{p\in B_{F}^{c}} c_{p}^{\ast}c_{p} =3 \mathcal{N}_{E}, \end{align} $$

which implies the desired claim.

From these results and equation (4.34), we immediately obtain the following:

Proposition 4.6. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,\left(Q_{1}^{k}(A)-\operatorname{\mathrm{tr}}(A)\right)\Psi\right\rangle \right|\leq5\left\Vert A\right\Vert _{\operatorname{\mathrm{Op}}}\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align*}$$

Next, we turn to $Q_{2}^{k}(B)$ .

Proposition 4.7. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,Q_{2}^{k}(B)\Psi\right\rangle \right|\leq2\left\Vert B\right\Vert _{\operatorname{\mathrm{HS}}}\sqrt{\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle }\leq2\left\Vert B\right\Vert _{\operatorname{\mathrm{HS}}}\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle. \end{align*}$$

Proof. We have (using that the $b_{k}$ operators commute)

(4.40) $$ \begin{align} \left\langle \Psi,Q_{2}^{k}(B)\Psi\right\rangle &=\sum_{p\in L_{k}^{\pm}}\left\langle \Psi,\left(b_{k}^{\ast}\left(Be_{p}\right)b_{k}^{\ast}(e_{p})+b_{k}(e_{p})b_{k}\left(Be_{p}\right)\right)\Psi\right\rangle \nonumber\\ &=2\sum_{p\in L_{k}^{\pm}}\mathrm{Re} \left\langle b_{k}^{\ast}\left(Be_{p}\right)\Psi,b_{k}(e_{p})\Psi\right\rangle, \end{align} $$

so using the estimates of Proposition 4.2 and the Cauchy-Schwarz inequality, we conclude that

(4.41) $$ \begin{align} \left|\left\langle \Psi,Q_{2}^{k}(B)\Psi\right\rangle \right| &\leq 2\sum_{p\in L_{k}^{\pm}}\left\Vert b_{k}^{\ast}\left(Be_{p}\right)\Psi\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert \leq2\sqrt{\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle }\sum_{p\in L_{k}^{\pm}}\left\Vert Be_{p}\right\Vert \left\Vert b_{\overline{k,p}}\Psi\right\Vert \nonumber\\ & \leq2\sqrt{\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle }\sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert Be_{p}\right\Vert ^{2}}\sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}} \nonumber \\ &\leq2\left\Vert B\right\Vert _{\text{HS}}\left\langle \Psi,\left(1+\mathcal{N}_{E}\right)\Psi\right\rangle, \end{align} $$

where we again used that $\left\Vert b_{\overline {k,p}}\Psi \right\Vert \leq \left\Vert c_{p}\Psi \right\Vert $ .

Kinetic estimates for quadratic operators

Finally, let us improve the estimates in this subsection by using the kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ instead of the number operator $\mathcal {N}_E$ .

Proposition 4.8. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,\tilde{Q}_{1}^{k}(A)\Psi\right\rangle \right|\leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}A\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{Op}}}\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle. \end{align*}$$

Proof. Let $(x_{i})_{i}$ be an eigenbasis for $(h_{k}^{\oplus })^{-\frac {1}{2}}A(h_{k}^{\oplus })^{-\frac {1}{2}}$ with eigenvalues $\left(\mu _{i}\right)_i$ . By Lemma 3.3, we then see that we may write $\tilde {Q}_{1}^{k}(A)$ as

(4.42) $$ \begin{align} \tilde{Q}_{1}^{k}(A) & =\sum_{i}b_{k}^{\ast}\left(Ax_{i}\right)b_{k}(x_{i})=\sum_{i}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}A\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)b_{k}(x_{i})\nonumber \\ & =\sum_{i}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}A\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}x_{i}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)\\ & =\sum_{i}\mu_{i}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right),\nonumber \end{align} $$

and so we can estimate

(4.43) $$ \begin{align} \left|\left\langle \Psi,\tilde{Q}_{1}^{k}(A)\Psi\right\rangle \right| &\leq\left(\max_{1\leq i\leq\left|L_{k}^{\pm}\right|}\left|\mu_{i}\right|\right)\sum_{i}\left\langle \Psi,b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)\Psi\right\rangle \\ & =\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}A\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}}\left\langle \Psi,\sum_{i}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)\Psi\right\rangle .\nonumber \end{align} $$

Applying Lemma 3.3 again, we also see that

(4.44) $$ \begin{align} \sum_{i}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}x_{i}\right) &=\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}e_{p}\right)b_{k}\left(\left(h_{k}^{\oplus}\right)^{\frac{1}{2}}e_{p}\right)\nonumber \\ &=\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}b_{\overline{k,p}}^{\ast}b_{\overline{k,p}}, \end{align} $$

so by equation (4.32), we obtain the desired bound of

(4.45) $$ \begin{align} \left|\left\langle \Psi,\tilde{Q}_{1}^{k}(A)\Psi\right\rangle \right|\leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}A\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}}\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle. \end{align} $$

Next are the $\varepsilon _{k}(A)$ terms. These we cannot estimate in terms of $H_{\operatorname {\mathrm {kin}}}^{\prime }$ , but for A of diagonal form, we can still control them strongly:

Proposition 4.9. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $A^{\oplus }=\left(\begin {array}{cc} A & 0\\ 0 & A \end {array}\right):\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,\varepsilon_{k}\left(A^{\oplus}\right)\Psi\right\rangle \right|\leq3\left(\max_{p\in L_{k}}\left|\left\langle e_{p},Ae_{p}\right\rangle \right|\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align*}$$

Proof. By the assumed form of $A^{\oplus }$ , we may write $\varepsilon _{k}\left(A^{\oplus }\right)$ as

(4.46) $$ \begin{align} \varepsilon_{k}\left(A^{\oplus}\right) & =\sum_{p\in L_{k}^{\pm}}\varepsilon_{k,k}\left(e_{p};A^{\oplus}e_{p}\right)=\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{q},A^{\oplus}e_{p}\right\rangle \varepsilon_{k,k}\left(\overline{k,p};\overline{k,q}\right)\nonumber \\ & =-\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{q},A^{\oplus}e_{p}\right\rangle \left(\delta_{p,q}c_{\overline{q-k}}c_{\overline{p-k}}^{\ast}+\delta_{\overline{p-k},\overline{q-k}}c_{q}^{\ast}c_{p}\right)\nonumber \\ & =-\sum_{p,q\in L_{k}}\left\langle e_{q},Ae_{p}\right\rangle \left(\delta_{p,q}c_{q-k}c_{p-k}^{\ast}+\delta_{p-k,q-k}c_{q}^{\ast}c_{p}\right)\\ &\quad -\sum_{p,q\in L_{-k}}\left\langle e_{-q},Ae_{-p}\right\rangle \left(\delta_{p,q}c_{q+k}c_{p+k}^{\ast}+\delta_{p+k,q+k}c_{q}^{\ast}c_{p}\right)\nonumber \\ & =-\sum_{p\in L_{k}}\left\langle e_{p},Ae_{p}\right\rangle \left(c_{p-k}c_{p-k}^{\ast}+c_{p}^{\ast}c_{p}\right)-\sum_{p\in L_{-k}}\left\langle e_{-p},Ae_{-p}\right\rangle \left(c_{p+k}c_{p+k}^{\ast}+c_{p}^{\ast}c_{p}\right)\nonumber \end{align} $$

since the terms with $p\in L_{k},q\in L_{-k}$ or $p\in L_{-k},q\in L_{k}$ vanish (because $L_k\cap L_{-k}=\emptyset $ and there are $\delta _{p,q}$ , $\delta _{p-k,q-k}$ in the summand). We can thus estimate

(4.47) $$ \begin{align} \left|\left\langle \Psi,\varepsilon_{k}\left(A^{\oplus}\right)\Psi\right\rangle \right| & \leq\sum_{p\in L_{k}}\left|\left\langle e_{p},Ae_{p}\right\rangle \right|\left\langle \Psi,\left(c_{p-k}c_{p-k}^{\ast}+c_{p}^{\ast}c_{p}\right)\Psi\right\rangle \nonumber \\ &\quad +\sum_{p\in L_{-k}}\left|\left\langle e_{-p},Ae_{-p}\right\rangle \right|\left\langle \Psi,\left(c_{p+k}c_{p+k}^{\ast}+c_{p}^{\ast}c_{p}\right)\Psi\right\rangle \\ & \leq\left(\max_{p\in L_{k}}\left|\left\langle e_{p},Ae_{p}\right\rangle \right|\right)\left\langle \Psi,\left(\sum_{p\in L_{k}^{\pm}}c_{p}^{\ast}c_{p}+\sum_{p\in L_{k}-k}c_{p}c_{p}^{\ast}+\sum_{p\in L_{-k}+k}c_{p}c_{p}^{\ast}\right)\Psi\right\rangle \nonumber \\ & \leq3\left(\max_{p\in L_{k}}\left|\left\langle e_{p},Ae_{p}\right\rangle \right|\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle .\nonumber \end{align} $$

Lastly, we consider the $Q_{2}^{k}(B)$ terms:

Proposition 4.10. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,Q_{2}^{k}(B)\Psi\right\rangle \right|\leq2\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}B\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}}\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle +2\left\Vert B\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}}\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\left\Vert \Psi\right\Vert. \end{align*}$$

Proof. By the Cauchy-Schwarz inequality and Proposition 4.4, we have

(4.48) $$ \begin{align} & \left|\left\langle \Psi,Q_{2}^{k}(B)\Psi\right\rangle \right| =\left|2\sum_{p\in L_{k}^{\pm}}\mathrm{ Re}\left(\left\langle \Psi,b_{k}\left(Be_{p}\right)b_{k}(e_{p})\Psi\right\rangle \right)\right|\leq2\sum_{p\in L_{k}^{\pm}}\left\Vert b_{k}^{\ast}\left(Be_{p}\right)\Psi\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert \nonumber \\ & \leq2\sum_{p\in L_{k}^{\pm}}\left(\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Be_{p}\right\Vert \sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }+\left\Vert Be_{p}\right\Vert \left\Vert \Psi\right\Vert \right)\left\Vert b_{k}(e_{p})\Psi\right\Vert \\ & \leq2\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\sum_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Be_{p}\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert +2\left\Vert \Psi\right\Vert \sum_{p\in L_{k}^{\pm}}\left\Vert Be_{p}\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert .\nonumber \end{align} $$

For the first sum, we can again apply the Cauchy-Schwarz inequality and (4.32):

(4.49) $$ \begin{align} \sum_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Be_{p}\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert & \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}^{-1}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Be_{p}\right\Vert ^{2}}\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}}\\ & \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}B\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}e_{p}\right\Vert ^{2}}\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle },\nonumber \end{align} $$

and we likewise estimate the second sum as

(4.50) $$ \begin{align} \sum_{p\in L_{k}^{\pm}}\left\Vert Be_{p}\right\Vert \left\Vert b_{k}(e_{p})\Psi\right\Vert & \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}^{-1}\left\Vert Be_{p}\right\Vert ^{2}}\sqrt{\sum_{p\in L_{k}^{\pm}}\lambda_{\overline{k,p}}\left\Vert b_{\overline{k,p}}\Psi\right\Vert ^{2}}\\ & \leq\sqrt{\sum_{p\in L_{k}^{\pm}}\left\Vert B\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}e_{p}\right\Vert ^{2}}\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }.\nonumber \end{align} $$

The claim now follows by recognizing the Hilbert-Schmidt norms.

5 The quasi-bosonic Bogolubov transformation

Now we are prepared to define the quasi-bosonic Bogolubov transformation that will approximately diagonalize the Hamiltonian in (4.13),

(5.1) $$ \begin{align} \sum_{k\in S_C} ( Q_1^k (h_k^{\oplus} + A_k^{\oplus}) + Q_2^k (B_k^{\oplus}) - 2\, \mathrm{tr}(h_k)), \end{align} $$

where $h_k^{\oplus }$ , $A_k^{\oplus }$ , $B_k^{\oplus }$ are defined in (4.11) and (4.7).

We define the generator $\mathcal {K}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ of the Bogolubov transformation as follows. Let $(K_{k}^{\oplus })_{k\in S_{C}}$ be a collection of symmetric operators $K_{k}^{\oplus }:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ . Then we define

(5.2) $$ \begin{align} \mathcal{K} & =\frac{1}{2}\sum_{k\in S_{C}}\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{p},K_{k}^{\oplus}e_{q}\right\rangle \left(b_{\overline{k,p}}b_{\overline{k,q}}-b_{\overline{k,q}}^{\ast}b_{\overline{k,p}}^{\ast}\right)\\ & =\frac{1}{2}\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}\left(K_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})-b_{k}^{\ast}(e_{p})b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right)\right)\nonumber \end{align} $$

in analogy with equation (3.9). As in the bosonic case, $\mathcal {K}$ is seen to be a skew-symmetric operator. ⁸ Moreover, unlike the bosonic case, $\mathcal {K}$ is now a bounded operator by the same argument that $Q_{1}^{k}(\cdot )$ and $Q_{2}^{k}(\cdot )$ are. Therefore, $\mathcal {K}$ generates a unitary transformation $e^{\mathcal {K}}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ , which is the quasi-bosonic Bogolubov transformation.

The specific kernels $K_k^\oplus $ we will use are those which diagonalize the corresponding bosonic Hamiltonian exactly, but first we will consider the action of $e^{\mathcal {K}}$ on quadratic operators and the localized kinetic operator more generally.

5.1 Transformation of quadratic operators

By exploiting the similarity of our quasi-bosonic definitions with the exact bosonic case, we can now easily deduce the analogues of Propositions 3.2 and 3.4:

Proposition 5.1. For all $k\in S_{C}$ , $\varphi \in \ell ^{2}(L_{k}^{\pm })$ and symmetric operators $(K_{l}^{\oplus })_{l\in S_{C}}$ , it holds that

$$ \begin{align*} \left[\mathcal{K},b_{k}(\varphi )\right] & =b_{k}^{\ast}\left(K_{k}^{\oplus}\varphi\right)+\mathcal{E}_{k}(\varphi ),\\ \left[\mathcal{K},b_{k}^{\ast}(\varphi )\right] & =b_{k}\left(K_{k}^{\oplus}\varphi\right)+\mathcal{E}_{k}(\varphi )^{\ast}, \end{align*} $$

where

$$\begin{align*}\mathcal{E}_{k}(\varphi )=\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}\left(\varphi;e_{q}\right)\right\}. \end{align*}$$

Proof. We calculate using the commutation relations of (4.15) that

(5.3) $$ \begin{align} \left[\mathcal{K},b_{k}(\varphi )\right] & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(\left[b_{l}\left(K_{l}^{\oplus}e_{q}\right)b_{l}(e_{p}),b_{k}(\varphi )\right]-\left[b_{l}^{\ast}\left(e_{q}\right)b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),b_{k}(\varphi )\right]\right)\nonumber \\ & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(b_{l}^{\ast}\left(e_{q}\right)\left[b_{k}(\varphi ),b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\right]+\left[b_{k}(\varphi ),b_{l}^{\ast}\left(e_{q}\right)\right]b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\right)\nonumber \\ & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}b_{l}^{\ast}\left(e_{q}\right)\left(\delta_{k,l}\left\langle \varphi,K_{l}^{\oplus}e_{q}\right\rangle +\varepsilon_{k,l}\left(\varphi;K_{l}^{\oplus}e_{q}\right)\right)\nonumber\\ &\quad +\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(\delta_{k,l}\left\langle \varphi,e_{q}\right\rangle +\varepsilon_{k,l}\left(\varphi;e_{q}\right)\right)b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\nonumber \\ & =\frac{1}{2}b_{k}^{\ast}\left(\sum_{q\in L_{k}^{\pm}}\left\langle \varphi,K_{k}^{\oplus}e_{q}\right\rangle e_{q}\right)+\frac{1}{2}b_{k}^{\ast}\left(K_{k}^{\oplus}\sum_{q\in L_{k}^{\pm}}\left\langle \varphi,e_{q}\right\rangle e_{q}\right)+\mathcal{E}_{k}(\varphi )\nonumber \\ & =b_{k}^{\ast}\left(K_{k}^{\oplus}\varphi\right)+\mathcal{E}_{k}(\varphi ) \end{align} $$

for $\mathcal {E}_{k}(\varphi )$ given by

(5.4) $$ \begin{align} \mathcal{E}_{k}(\varphi ) & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(b_{l}^{\ast}\left(e_{q}\right)\varepsilon_{k,l}\left(\varphi;K_{l}^{\oplus}e_{q}\right)+\varepsilon_{k,l}\left(\varphi;e_{q}\right)b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\right)\nonumber\\ & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}\left(\varphi;e_{q}\right)\right\} , \end{align} $$

where we used Lemma 3.3 to simplify the expression (as $x,y\mapsto b_{l}^{\ast } (x )\varepsilon _{k,l}\left(\varphi ;y\right)$ is bilinear for fixed $\varphi $ and $K_{k}^{\oplus }$ is symmetric). The commutator $\left[\mathcal {K},b_{k}^{\ast }(\varphi )\right]$ follows by taking the adjoint.

From this, we easily deduce the commutator of $\mathcal {K}$ with quadratic operators:

Proposition 5.2. For all $k\in S_{C}$ and symmetric operators $A,B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , it holds that

$$ \begin{align*} \left[\mathcal{K},Q_{1}^{k}(A)\right] & =Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},A\right\} \right)+\mathcal{E}_{1}^{k}(A)\\ \left[\mathcal{K},Q_{2}^{k}(B)\right] & =Q_{1}^{k}\left(\left\{ K_{k}^{\oplus},B\right\} \right)+\mathcal{E}_{2}^{k}(B), \end{align*} $$

where

$$ \begin{align*} \mathcal{E}_{1}^{k}(A) & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\left(\left\{ b_{k}^{\ast}(Ae_{p}),\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}(e_{p};e_{q})\right\} \right\} \right.\\ & \quad\left.+\left\{ \left\{ \varepsilon_{l,k}\left(e_{q};e_{p}\right),b_{l}\left(K_{l}^{\oplus}e_{q}\right)\right\} ,b_{k}(Ae_{p})\right\} \right)\\ \mathcal{E}_{2}^{k}(B) & =\frac{1}{2}\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\left(\left\{ b_{k}^{\ast}\left(Be_{p}\right),\left\{ b_{l}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{l,k}\left(e_{q};e_{p}\right)\right\} \right\} \right.\\ & \quad \left.+\left\{ \left\{ \varepsilon_{k,l}(e_{p};e_{q}),b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\right\} ,b_{k}\left(Be_{p}\right)\right\} \right). \end{align*} $$

Proof. We compute using the commutators of the previous proposition (and Lemma 3.3, to simplify the resulting expressions) that

(5.5) $$ \begin{align} \left[\mathcal{K},Q_{1}^{k}(A)\right] & =\sum_{p\in L_{k}^{\pm}}\left(\left[\mathcal{K},b_{k}^{\ast}(Ae_{p})b_{k}(e_{p})\right]+\left[\mathcal{K},b_{k}(e_{p})b_{k}^{\ast}(Ae_{p})\right]\right)\nonumber \\ & =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\left[\mathcal{K},b_{k}(e_{p})\right]+\left[\mathcal{K},b_{k}^{\ast}(Ae_{p})\right]b_{k}(e_{p})\right)\nonumber \\ &\quad +\sum_{p\in L_{k}^{\pm}}\left(b_{k}(e_{p})\left[\mathcal{K},b_{k}^{\ast}(Ae_{p})\right]+\left[\mathcal{K},b_{k}(e_{p})\right]b_{k}^{\ast}(Ae_{p})\right)\nonumber\\ & =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\left(b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right)+\mathcal{E}_{k}(e_{p})\right)+\left(b_{k}\left(K_{k}^{\oplus}Ae_{p}\right)+\mathcal{E}_{k}(Ae_{p})^{\ast}\right)b_{k}(e_{p})\right)\nonumber \\ &\quad +\sum_{p\in L_{k}^{\pm}}\left(b_{k}(e_{p})\left(b_{k}\left(K_{k}^{\oplus}Ae_{p}\right)+\mathcal{E}_{k}(Ae_{p})^{\ast}\right)+\left(b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right)+\mathcal{E}_{k}(e_{p})\right)b_{k}^{\ast}(Ae_{p})\right)\nonumber \\ & =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}\left(\left(AK_{k}^{\oplus}+K_{k}^{\oplus}A\right)e_{p}\right)b_{k}^{\ast}(e_{p})+b_{k}(e_{p})b_{k}\left(\left(K_{k}^{\oplus}A+K_{k}^{\oplus}A\right)e_{p}\right)\right)\nonumber \\ &\quad +\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\mathcal{E}_{k}(e_{p})+\mathcal{E}_{k}(e_{p})^{\ast}b_{k}(Ae_{p})+b_{k}(Ae_{p})\mathcal{E}_{k}(e_{p})^{\ast}+\mathcal{E}_{k}(e_{p})b_{k}^{\ast}(Ae_{p})\right)\nonumber \\ & =Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},A\right\} \right)+\sum_{p\in L_{k}^{\pm}}\left(\left\{ b_{k}^{\ast}(Ae_{p}),\mathcal{E}_{k}(e_{p})\right\} +\left\{ \mathcal{E}_{k}(e_{p})^{\ast},b_{k}(Ae_{p})\right\} \right) \end{align} $$

and

(5.6) $$ \begin{align} & \sum_{p\in L_{k}^{\pm}}\left(\left\{ b_{k}^{\ast}(Ae_{p}),\mathcal{E}_{k}(e_{p})\right\} +\left\{ \mathcal{E}_{k}(e_{p})^{\ast},b_{k}(Ae_{p})\right\} \right)\nonumber \\ &\quad =\frac{1}{2}\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\left(\left\{ b_{k}^{\ast}(Ae_{p}),\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}(e_{p};e_{q})\right\} \right\} \right.\\ & \qquad\left.+\left\{ \left\{ \varepsilon_{l,k}\left(e_{q};e_{p}\right),b_{l}\left(K_{l}^{\oplus}e_{q}\right)\right\} ,b_{k}(Ae_{p})\right\} \right)=\mathcal{E}_{1}^{k}(A) \nonumber \end{align} $$

as $\varepsilon _{k,l}(e_{p};e_{q})^{\ast }=\varepsilon _{l,k}\left(e_{q};e_{p}\right)$ . The computation of $Q_{2}^{k}(B)$ is similar.

Action of $e^{\mathcal {K}}$ on quadratic operators

With the commutators calculated, we are now ready to determine the full action of $e^{\mathcal {K}}$ on the quadratic operators $Q_{1}^{k}(\cdot )$ and $Q_{2}^{k}(\cdot )$ . Rather than appeal to the Baker-Campbell-Hausdorff formula, which would also require describing the commutators $\left[\mathcal {K},\mathcal {E}_{1}^{k}(A)\right]$ , etc., we will employ a ‘Duhamel-type’ argument which allows us to more selectively expand the operator $e^{\mathcal {K}}$ .

As in Section 3, we use the notation $\mathcal {A}_{K_{k}^{\oplus }}=\left\{ K_{k}^{\oplus },\cdot \right\}$ for anticommutators with $K_{k}^{\oplus }$ .

Before stating the proposition, we must make a remark. To use these identities, we will need to take limits, and to justify those limits, we need some general estimates on operators of the form $Q_{1}^{k}(\cdot ),Q_{2}^{k}(\cdot ),\mathcal {E}_{1}^{k}(\cdot ),\mathcal {E}_{2}^{k}(\cdot )$ . The Propositions 4.6, 4.7 establish these for $Q_{1}^{k}(\cdot )$ and $Q_{2}^{k}(\cdot )$ , while Proposition 6.4 will establish these for $\mathcal {E}_{1}^{k}(\cdot )$ and $\mathcal {E}_{2}^{k}(\cdot )$ .

The statement follows:

Proposition 5.3. For all $k\in S_{C}$ and symmetric $A,B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , it holds that

$$ \begin{align*} e^{\mathcal{K}}Q_{1}^{k}(A)e^{-\mathcal{K}} & =\frac{1}{2}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}Ae^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}Ae^{-K_{k}^{\oplus}}\right)+\frac{1}{2}Q_{2}^{k}\left(e^{K_{k}^{\oplus}}Ae^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}Ae^{-K_{k}^{\oplus}}\right)\\ &\quad +\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\cosh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)(A)\right)+\mathcal{E}_{2}^{k}\left(\sinh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)(A)\right)\right)e^{-t\mathcal{K}}\,dt\\ e^{\mathcal{K}}Q_{2}^{k}(B)e^{-\mathcal{K}} & =\frac{1}{2}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}Be^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}Be^{-K_{k}^{\oplus}}\right)+\frac{1}{2}Q_{2}^{k}\left(e^{K_{k}^{\oplus}}Be^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}Be^{-K_{k}^{\oplus}}\right)\\ &\quad +\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\sinh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)(B)\right)+\mathcal{E}_{2}^{k}\left(\cosh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)(B)\right)\right)e^{-t\mathcal{K}}\,dt, \end{align*} $$

the integrals being Riemann integrals of bounded operators.

Proof. We consider $e^{\mathcal {K}}Q_{1}^{k}(A)e^{-\mathcal {K}}$ , with the argument for $e^{\mathcal {K}}Q_{2}^{k}(B)e^{-\mathcal {K}}$ being similar. We first claim that for any $n\in \mathbb {N}$ ,

(5.7) $$ \begin{align} e^{\mathcal{K}}Q_{1}^{k}(A)e^{-\mathcal{K}} &=Q_{1}^{k}\left(\sum_{m=0}^{n_{1}}\frac{1}{\left(2m\right)!}\mathcal{A}_{K_{k}^{\oplus}}^{2m}(A)\right)+Q_{2}^{k}\left(\sum_{m=0}^{n_{2}}\frac{1}{\left(2m+1\right)!}\mathcal{A}_{K_{k}^{\oplus}}^{2m+1}(A)\right)\\ &\quad +\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\sum_{m=0}^{n_{1}}\frac{1}{\left(2m\right)!}\mathcal{A}_{(1-t)K_{k}^{\oplus}}^{2m}(A)\right)+\mathcal{E}_{2}^{k}\left(\sum_{m=0}^{n_{2}}\frac{1}{\left(2m+1\right)!}\mathcal{A}_{(1-t)K_{k}^{\oplus}}^{2m+1}(A)\right)\right)e^{-t\mathcal{K}}\,dt\nonumber \\ &\quad +\frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{K}}Q_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{K}}(1-t)^{n-1}\,dt,\nonumber \end{align} $$

where, for brevity, $\overline {n-1}=n-1\ \mod 2$ and $n_{1},n_{2}$ are the largest integers such that $2n_{1}<n$ and $2n_{2}+1<n$ , respectively.

We proceed by induction. For $n=1$ , we find by the fundamental theorem of calculus that

(5.8) $$ \begin{align} e^{\mathcal{K}}Q_{1}^{k}(A)e^{-\mathcal{K}} & =Q_{1}^{k}(A)+\int_{0}^{1}\frac{d}{dt}\left(e^{t\mathcal{K}}Q_{1}^{k}(A)e^{-t\mathcal{K}}\right)\,dt=Q_{1}^{k}(A)+\int_{0}^{1}e^{t\mathcal{K}}\left[\mathcal{K},Q_{1}^{k}(A)\right]e^{-t\mathcal{K}}\,dt\nonumber \\ & =Q_{1}^{k}(A)+\int_{0}^{1}e^{t\mathcal{K}}\left(Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},A\right\} \right)+\mathcal{E}_{1}^{k}(A)\right)e^{-t\mathcal{K}}\,dt\\ & =Q_{1}^{k}(A)+\int_{0}^{1}e^{t\mathcal{K}}\mathcal{E}_{1}^{k}(A)e^{-t\mathcal{K}}\,dt+\int_{0}^{1}e^{t\mathcal{K}}Q_{2}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}(A)\right)e^{-t\mathcal{K}}\,dt\nonumber \end{align} $$

by the commutator of Proposition 5.2, which is the statement for $n=1$ (in this case, $n_1=0$ and $n_2=-1$ , so $\sum _{m=0}^{n_1}$ contains one term and $\sum _{m=0}^{n_2}$ is empty).

For the inductive step, we now assume that case n holds. Integrating the last term of equation (5.7) by parts, we find that

(5.9) $$ \begin{align} & \frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{K}}Q_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{K}}(1-t)^{n-1}dt\nonumber \\ &\quad =\frac{1}{(n-1)!}\left[e^{t\mathcal{K}}Q_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{K}}\left(-\frac{(1-t)^{n}}{n}\right)\right]_{0}^{1}\nonumber \\ &\qquad -\frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{K}}\left[\mathcal{K},Q_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)\right]e^{-t\mathcal{K}}\left(-\frac{(1-t)^{n}}{n}\right)dt\\ &\quad =\frac{1}{n!}Q_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)+\frac{1}{n!}\int_{0}^{1}e^{t\mathcal{K}}\left(Q_{\overline{n}}^{k}\left(\left\{ K_{k}^{\oplus},\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right\} \right)+\mathcal{E}_{\overline{n-1}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)\right)e^{-t\mathcal{K}}(1-t)^{n}dt\nonumber \\ &\quad =Q_{\overline{n-1}}^{k}\left(\frac{1}{n!}\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\right)+\int_{0}^{1}e^{t\mathcal{K}}\mathcal{E}_{\overline{n-1}}^{k}\left(\frac{1}{n!}\mathcal{A}_{(1-t)K_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{K}}\,dt\nonumber \\ &\qquad +\frac{1}{n!}\int_{0}^{1}e^{t\mathcal{K}}Q_{\overline{n}}^{k}\left(\mathcal{A}_{K_{k}^{\oplus}}^{n+1}(A)\right)e^{-t\mathcal{K}}(1-t)^{n}dt,\nonumber \end{align} $$

where we also used that

(5.10) $$ \begin{align} (1-t)^{n}\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)=\big((1-t)\mathcal{A}_{K_{k}^{\oplus}}(A)\big)^{n}=\mathcal{A}_{(1-t)K_{k}^{\oplus}}^{n}(A). \end{align} $$

Inserting this into (5.7) and collecting like terms yields the statement for case $n+1$ .

We now deduce the statement from (5.7) by taking $n\to \infty $ . Recall the identities

(5.11) $$ \begin{align} \cosh\big(\mathcal{A}_{K_{k}^{\oplus}}\big)(T) & =\frac{1}{2}\big(e^{K_{k}^{\oplus}}Te^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}Te^{-K_{k}^{\oplus}}\big) \\ \sinh\big(\mathcal{A}_{K_{k}^{\oplus}}\big)(T) & =\frac{1}{2}\big(e^{K_{k}^{\oplus}}Te^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}Te^{-K_{k}^{\oplus}}\big)\nonumber \end{align} $$

from Proposition 3.5 and note that $\left((n-1)!\right)^{-1}\mathcal {A}_{K_{k}^{\oplus }}^{n}(A)\to 0$ as $n\to \infty $ . By Proposition 4.6,

$$ \begin{align*}Q_{1}^{k}\left(\sum_{m=0}^{n_{1}}\frac{1}{\left(2m\right)!}\mathcal{A}_{K_{k}^{\oplus}}^{2m}(A)\right)\to \frac{1}{2}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}Ae^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}Ae^{-K_{k}^{\oplus}}\right) \end{align*} $$

and

$$ \begin{align*}\frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{K}}Q_{1}^{k}\big(\mathcal{A}_{K_{k}^{\oplus}}^{n}(A)\big)e^{-t\mathcal{K}}(1-t)^{n-1}\,dt \to 0. \end{align*} $$

Similar convergence for $Q_2$ is justified by Proposition 4.7. The convergence for $\mathcal {E}_{1}^{k}$ and $\mathcal {E}_{2}^{k}$ follows from Proposition 6.4.

Remark on the transformation of excitation operators

Let us make a quick remark on why we choose to approach the Bogolubov transformation from the point of view of quadratic operators rather than the usual creation and annihilation operator approach. Recall that in the exact bosonic case the creation and annihilation operators transformed under a Bogolubov transformation as

(5.12) $$ \begin{align} e^{\mathcal{K}}a(\varphi )e^{-\mathcal{K}} & =a\left(\cosh\left(K\right)\varphi\right)+a^{\ast}\left(\sinh\left(K\right)\varphi\right)\\ e^{\mathcal{K}}a^{\ast}(\varphi )e^{-\mathcal{K}} & =a^{\ast}\left(\cosh\left(K\right)\varphi\right)+a\left(\sinh\left(K\right)\varphi\right).\nonumber \end{align} $$

In the quasi-bosonic setting, we can use the commutators of Proposition 5.1 and a similar Duhamel-type argument to what we just applied to conclude that

(5.13) $$ \begin{align} e^{\mathcal{K}}b_{k}(\varphi )e^{-\mathcal{K}} & =b_{k}\left(\cosh\left(K_{k}^{\oplus}\right)\varphi\right)+b_{k}^{\ast}\left(\sinh\left(K_{k}^{\oplus}\right)\varphi\right)\\ &\quad +\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{k}\left(\cosh\left((1-t)K_{k}^{\oplus}\right)\varphi\right)+\mathcal{E}_{k}\left(\sinh\left((1-t)K_{k}^{\oplus}\right)\varphi\right)^{\ast}\right)e^{-t\mathcal{K}}\,dt\nonumber \end{align} $$

with a similar expression for $e^{\mathcal {K}}b_{k}^{\ast }(\varphi )e^{-\mathcal {K}}$ . This is a more cumbersome expression to work with, and if we were to describe $e^{\mathcal {K}}Q_{1}^{k}(A)e^{-\mathcal {K}}$ by transforming the individual terms of $Q_{1}^{k}(A)$ like this rather than transforming $Q_{1}^{k}(A)$ as a whole, the error terms would not only go from being under a single integral to involving the product of two integrals, it would also involve cross terms between the bosonic terms and the error terms of equation (5.13). These cross terms, in particular, would severely reduce the quality of the final error estimate. Hence, we prefer the quadratic operator approach in the quasi-bosonic setting.

5.2 Transformation of the kinetic operator

There remains the task of describing the action of $e^{\mathcal {K}}$ on the localized kinetic operator $H_{\operatorname {\mathrm {kin}}}^{\prime }$ . For this, we must first formulate $H_{\operatorname {\mathrm {kin}}}^{\prime }$ – or rather the commutator $[H_{\operatorname {\mathrm {kin}}}^{\prime },b_{k,p}^{\ast }]$ calculated in (1.74) – within the general framework that we have introduced in this section. Recalling the operators $h_{k}^{\oplus }:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ in (4.11), then by (1.74) and linearity it follows that

(5.14) $$ \begin{align} \left[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{k}(\varphi )\right]=-2\,b_{k}\left(h_{k}^{\oplus}\varphi\right),\quad\left[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{k}^{\ast}(\varphi )\right]=2\,b_{k}^{\ast}\left(h_{k}^{\oplus}\varphi\right) \end{align} $$

for all $\varphi \in \ell ^{2}(L_{k}^{\pm })$ . (The factor of $2$ is introduced here because in the analogy of equation (4.9), $H_{\operatorname {\mathrm {kin}}}^{\prime }$ appears like a $\text {d}\Gamma (\cdot )=\frac {1}{2}Q_{1}(\cdot )-\frac {1}{2}\text {tr}(\cdot )$ term rather than a pure $Q_{1}(\cdot )$ term.)

We now calculate $\left[\mathcal {K},H_{\operatorname {\mathrm {kin}}}^{\prime }\right]$ as follows:

Proposition 5.4. $H_{\operatorname {\mathrm {kin}}}^{\prime }$ obeys

$$\begin{align*}\left[\mathcal{K},H_{\operatorname{\mathrm{kin}}}^{\prime}\right]=\sum_{k\in S_{C}}Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right). \end{align*}$$

Proof. We compute, using the commutators of equation (5.14) and Lemma 3.3, that

(5.15) $$ \begin{align} \left[\mathcal{K},H_{\operatorname{\mathrm{kin}}}^{\prime}\right] & =\frac{1}{2}\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(\left[b_{k}\left(K_{k}^{\oplus}e_{p}\right)b_{k}(e_{p}),H_{\operatorname{\mathrm{kin}}}^{\prime}\right]-\left[b_{k}^{\ast}(e_{p})b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right),H_{\operatorname{\mathrm{kin}}}^{\prime}\right]\right)\nonumber \\ & =\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(e_{p})b_{k}^{\ast}\left(h_{k}^{\oplus}K_{k}^{\oplus}e_{p}\right)+b_{k}^{\ast}\left(h_{k}^{\oplus}e_{p}\right)b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right)\right)\\ &\quad +\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}\left(K_{k}^{\oplus}e_{p}\right)b_{k}\left(h_{k}^{\oplus}e_{p}\right)+b_{k}\left(h_{k}^{\oplus}K_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})\right)\nonumber \\ & =\sum_{k\in S_{C}}Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right).\nonumber \end{align} $$

Note that because the commutator $\left[H_{\operatorname {\mathrm {kin}}}^{\prime },b_{k}^{\ast }(\varphi )\right]=2\,b_{k}^{\ast }\left(h_{k}^{\oplus }\varphi \right)$ exactly mirrors the bosonic case (in that there is no additional error term), the commutator $\left[\mathcal {K},H_{\operatorname {\mathrm {kin}}}^{\prime }\right]$ is likewise ‘purely bosonic’, being simply a sum of $Q_{2}^{k}(\cdot )$ terms without error terms such as those appearing in the statement of Proposition 5.2. With the groundwork laid, we can now easily deduce the following:

Proposition 5.5. $H_{\operatorname {\mathrm {kin}}}^{\prime }$ obeys

$$ \begin{align*} & e^{\mathcal{K}}H_{\operatorname{\mathrm{kin}}}^{\prime}e^{-\mathcal{K}}=H_{\operatorname{\mathrm{kin}}}^{\prime}\\ & +\sum_{k\in S_{C}}\left(\frac{1}{2}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}h_{k}^{\oplus}e^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}h_{k}^{\oplus}e^{-K_{k}^{\oplus}}-2h_{k}^{\oplus}\right)+\frac{1}{2}Q_{2}^{k}\left(e^{K_{k}^{\oplus}}h_{k}^{\oplus}e^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}h_{k}^{\oplus}e^{-K_{k}^{\oplus}}\right)\right)\\ & +\sum_{k\in S_{C}}\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\cosh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}\right)-h_{k}^{\oplus}\right)+\mathcal{E}_{2}^{k}\left(\sinh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}\right)\right)\right)e^{-t\mathcal{K}}\,dt. \end{align*} $$

Proof. By adding and subtracting, we have

(5.16) $$ \begin{align} e^{\mathcal{K}}H_{\operatorname{\mathrm{kin}}}^{\prime}e^{-\mathcal{K}}=\sum_{k\in S_{C}}e^{\mathcal{K}}Q_{1}^{k}\left(h_{k}^{\oplus}\right)e^{-\mathcal{K}}+e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}-\sum_{k\in S_{C}}Q_{1}^{k}\left(h_{k}^{\oplus}\right)\right)e^{-\mathcal{K}}, \end{align} $$

and the first term on the right-hand side is by Proposition 5.3,

(5.17) $$ \begin{align} &\sum_{k\in S_{C}}e^{\mathcal{K}}Q_{1}^{k}\left(h_{k}^{\oplus}\right)e^{-\mathcal{K}} \\ &\quad =\sum_{k\in S_{C}}\left(\frac{1}{2}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}h_{k}^{\oplus}e^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}h_{k}^{\oplus}e^{-K_{k}^{\oplus}}\right)+\frac{1}{2}Q_{2}^{k}\left(e^{K_{k}^{\oplus}}h_{k}^{\oplus}e^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}h_{k}^{\oplus}e^{-K_{k}^{\oplus}}\right)\right) \nonumber\\ &\qquad +\sum_{k\in S_{C}}\int_{0}^{1}e^{t\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\cosh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}\right)\right)+\mathcal{E}_{2}^{k}\left(\sinh\left(\mathcal{A}_{(1-t)K_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}\right)\right)\right)e^{-t\mathcal{K}}\,dt,\nonumber \end{align} $$

while the second is calculated using the commutators of the Propositions 5.2 and 5.4 to be

(5.18) $$ \begin{align} &e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}-\sum_{k\in S_{C}}Q_{1}^{k}\left(h_{k}^{\oplus}\right)\right)e^{-\mathcal{K}} - \left( H_{\operatorname{\mathrm{kin}}}^{\prime}-\sum_{k\in S_{C}}Q_{1}^{k}\left(h_{k}^{\oplus}\right) \right) \nonumber\\ &\quad =\int_{0}^{1}e^{t\mathcal{K}}\left[\mathcal{K},H_{\operatorname{\mathrm{kin}}}^{\prime}-\sum_{k\in S_{C}}Q_{1}^{k}\left(h_{k}^{\oplus}\right)\right]e^{-t\mathcal{K}}\,dt = -\sum_{k\in S_{C}}\int_{0}^{1}e^{t\mathcal{K}}\mathcal{E}_{1}^{k}\left(h_{k}^{\oplus}\right)e^{-t\mathcal{K}}\,dt , \end{align} $$

which yields the claim.

5.3 Fixing the transformation kernels

With all the transformation identities determined, we now choose the transformation kernels $(K_{k}^{\oplus })_{k\in S_{C}}$ such that $H_{\operatorname {\mathrm {kin}}}^{\prime }+\sum _{k\in S_{C}}H_{\text {int}}^{k}$ is diagonalized. For any choice of $(K_{k}^{\oplus })_{k\in S_{C}}$ , the Propositions 5.3 and 5.5 imply that

(5.19) $$ \begin{align} & e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}H_{\text{int}}^{k}\right)e^{-\mathcal{K}} \nonumber\\ &\quad =\frac{1}{2}\sum_{k\in S_{C}}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}}-2h_{k}^{\oplus}\right)\nonumber \\ &\qquad +\frac{1}{2}\sum_{k\in S_{C}}Q_{2}^{k}\left(e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}-e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}}\right)+H_{\operatorname{\mathrm{kin}}}^{\prime}+\text{error terms}. \end{align} $$

In analogy with the bosonic case, we consider this expression to be diagonalized provided the $Q_{2}^{k}(\cdot )$ terms vanish, whence the diagonalization condition is that

(5.20) $$ \begin{align} e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}=e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}}, \end{align} $$

which we note is the same as the diagonalization condition (equation (3.26)) of the exact bosonic quadratic Hamiltonian

(5.21) $$ \begin{align} H=Q_{1}\left(h_{k}^{\oplus}+A_{k}^{\oplus}\right)+Q_{2}\left(B_{k}^{\oplus}\right)\quad\text{on }\mathcal{F}^{+}\left(\ell^{2}\left(L_{k}^{\pm}\right)\right). \end{align} $$

Recalling the definitions of $h_k^{\oplus }$ , $A_k^{\oplus }$ and $B_k^{\oplus }$ from (4.11) and (4.7), we have

$$ \begin{align*} h_{k}^{\oplus}+A_{k}^{\oplus}\pm B_{k}^{\oplus} =\left(\begin{array}{cc} h_k+P_{v_k} & \pm P_{v_{k}}\\ \pm P_k & h_k + P_{v_{k}}\\ \end{array}\right)>0. \end{align*} $$

So by Theorem 3.1, the choice

$$ \begin{align*} K_{k}^{\oplus} & =-\frac{1}{2}\log\left(\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)^{-\frac{1}{2}}\left(\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)^{\frac{1}{2}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)^{\frac{1}{2}}\right)^{\frac{1}{2}}\right.\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;\left.\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)^{-\frac{1}{2}}\right) \end{align*} $$

is the unique diagonalizing kernel for the Hamiltonian. In this form, it is, however, not easy to see how $K_{k}^{\oplus }$ acts, so we will proceed slightly differently: we define $K_{k}^{\oplus }:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ by

(5.22) $$ \begin{align} K_{k}^{\oplus}=\left(\begin{array}{cc} 0 & K_{k}\\ K_{k} & 0 \end{array}\right), \end{align} $$

where the operator $K_{k}:\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ is given by

(5.23) $$ \begin{align} K_{k}=-\frac{1}{2}\log\left(h_{k}^{-\frac{1}{2}}\Big(h_{k}^{\frac{1}{2}}\left(h_{k}+2P_{v_{k}}\right)h_{k}^{\frac{1}{2}}\Big)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}\right)=-\frac{1}{2}\log\left(h_{k}^{-\frac{1}{2}}\Big(h_{k}^{2}+2P_{h_{k}^{\frac{1}{2}}v_{k}}\Big)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}\right). \end{align} $$

A kernel similar to $K_k$ also appeared in [5, 6]. Note that $K_{k}$ is precisely the diagonalizer of Theorem 3.1 for the exact bosonic quadratic Hamiltonian

(5.24) $$ \begin{align} H=Q_{1}\left(h_{k}^{\oplus}+P_{v_{k}}\right)+Q_{2}(P_{v_{k}})\quad\text{on }\mathcal{F}^{+}(\ell^{2}(L_{k})), \end{align} $$

rather than that of equation (5.21). Now we can verify that this $K_{k}^{\oplus }$ is, in fact, equal to the diagonalizing kernel:

Proposition 5.6. The operator $K_{k}^{\oplus }$ defined by the equations (5.22) and (5.23) satisfies

$$\begin{align*}e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}=e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}}=\left(\begin{array}{cc} E_{k} & 0\\ 0 & E_{k} \end{array}\right) \end{align*}$$

for $E_{k}=e^{-K_{k}}h_{k}e^{-K_{k}}$ .

Proof. It is easily verified that $e^{\pm K_{k}^{\oplus }}$ is given by

(5.25) $$ \begin{align} e^{\pm K_{k}^{\oplus}}=\left(\begin{array}{cc} \cosh\left(K_{k}\right) & \pm\sinh\left(K_{k}\right)\\ \pm\sinh\left(K_{k}\right) & \cosh\left(K_{k}\right) \end{array}\right), \end{align} $$

and so

(5.26) $$ \begin{align} & e^{\pm K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}\pm B_{k}^{\oplus}\right)e^{\pm K_{k}^{\oplus}}\nonumber\\ &\quad =\left(\begin{array}{cc} \cosh\left(K_{k}\right) &\quad \pm\sinh\left(K_{k}\right)\\ \pm\sinh\left(K_{k}\right) & \cosh\left(K_{k}\right) \end{array}\right)\left(\begin{array}{cc} h_{k}+P_{v_{k}} & \pm P_{v_{k}}\\ \pm P_{v_{k}} & h_{k}+P_{v_{k}} \end{array}\right)\left(\begin{array}{cc} \cosh\left(K_{k}\right) & \pm\sinh\left(K_{k}\right)\\ \pm\sinh\left(K_{k}\right) & \cosh\left(K_{k}\right) \end{array}\right)\nonumber \\ &\quad =\frac{1}{2}\left(\begin{array}{cc} e^{K_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{K_{k}}+e^{-K_{k}}h_{k}e^{-K_{k}} & \pm\left(e^{K_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{K_{k}}-e^{-K_{k}}h_{k}e^{-K_{k}}\right)\\ \pm\left(e^{K_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{K_{k}}-e^{-K_{k}}h_{k}e^{-K_{k}}\right) & e^{K_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{K_{k}}+e^{-K_{k}}h_{k}e^{-K_{k}} \end{array}\right). \end{align} $$

The condition

(5.27) $$ \begin{align} e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}=e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}} \end{align} $$

thus holds if and only if

(5.28) $$ \begin{align} e^{K_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{K_{k}}=e^{-K_{k}}h_{k}e^{-K_{k}}, \end{align} $$

which is the diagonalization condition for the bosonic Hamiltonian of equation (5.24). Theorem 3.1 asserts that this condition is satisfied for our choice of $K_{k}$ , and the claim follows.

5.4 Full transformation of the bosonizable terms

With the above choice of transformation kernels, we thus conclude that

(5.29) $$ \begin{align} & e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}H_{\text{int}}^{k}\right)e^{-\mathcal{K}}=H_{\operatorname{\mathrm{kin}}}^{\prime}+\text{error terms}\nonumber \\ &\qquad +\frac{1}{2}\sum_{k\in S_{C}}Q_{1}^{k}\left(e^{K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{K_{k}^{\oplus}}+e^{-K_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-K_{k}^{\oplus}}-2h_{k}^{\oplus}\right)\nonumber\\ &\quad =H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}Q_{1}^{k}\left(\begin{array}{cc} E_{k}-h_{k} & 0\\ 0 & E_{k}-h_{k} \end{array}\right)+\text{error terms}, \end{align} $$

and so we have succeeded in diagonalizing $H_{\operatorname {\mathrm {kin}}}^{\prime }+\sum _{k\in S_{C}}H_{\text {int}}^{k}$ while simultanously decoupling the spaces $\ell ^{2}\left(L_{\pm k}\right)\subset \ell ^{2}(L_{k}^{\pm })$ in a symmetric fashion. We still need to determine the exact form of the error terms, which we record in the following proposition:

Proposition 5.7. Let $S_{C}=\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ with $\gamma \in (0,1]$ . Then the unitary transformation $e^{\mathcal {K}}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ with $\mathcal {K}$ defined by (5.2), (5.22), (5.23) satisfies

$$ \begin{align*} & e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}H_{\operatorname{\mathrm{int}}}^{k}\right)e^{-\mathcal{K}}=H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}Q_{1}^{k} (E_k^{\oplus} - h_k^{\oplus})\\ &\quad +\sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{K}}\left(\mathcal{E}_{1}^{k}(A_k^\oplus(t))+\mathcal{E}_{2}^{k} (B_k^\oplus(t))\right)e^{-(1-t)\mathcal{K}}\,dt, \end{align*} $$

where $\mathcal {E}_1(\cdot )$ , $\mathcal {E}_2(\cdot )$ are defined in Proposition 5.2 and

$$ \begin{align*}E_k^{\oplus} - h_k^{\oplus}= \left(\begin{array}{cc} E_{k}-h_{k} & 0\\ 0 & E_{k}-h_{k} \end{array}\right),\,\, A_{k}^\oplus (t) = \left(\begin{array}{cc} A_{k} (t ) & 0\\ 0 & A_{k} (t ) \end{array}\right),\,\, B_{k}^\oplus (t) = \left(\begin{array}{cc} 0 & B_{k} (t )\\ B_{k} (t ) & 0 \end{array}\right) \end{align*} $$

with $E_k= e^{-K_k} h_k e^{-K_k}$ and the operators $A_{k} (t ),B_{k} (t ):\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ defined by

$$ \begin{align*} A_{k} (t ) & =\frac{1}{2}\left(e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}+e^{-tK_{k}}h_{k}e^{-tK_{k}}\right)-h_{k}\\ B_{k} (t ) & =\frac{1}{2}\left(e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}-e^{-tK_{k}}h_{k}e^{-tK_{k}}\right). \end{align*} $$

Proof. By the Propositions 5.3 and 5.5, the error terms are

$$ \begin{align*} & \sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{K}}\left(\mathcal{E}_{1}^{k}\left(\cosh\left(\mathcal{A}_{tK_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}+A_{k}^{\oplus}\right)+\sinh\left(\mathcal{A}_{tK_{k}^{\oplus}}\right)\left(B_{k}^{\oplus}\right)-h_{k}^{\oplus}\right)\right)e^{-(1-t)\mathcal{K}}\,dt\\ &\quad +\sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{K}}\left(\mathcal{E}_{2}^{k}\left(\sinh\left(\mathcal{A}_{tK_{k}^{\oplus}}\right)\left(h_{k}^{\oplus}+A_{k}^{\oplus}\right)+\cosh\left(\mathcal{A}_{tK_{k}^{\oplus}}\right)\left(B_{k}^{\oplus}\right)\right)\right)e^{-(1-t)\mathcal{K}}\,dt, \end{align*} $$

where we have reparametrized the integral by $t\mapsto 1-t$ to simplify the arguments of the $\mathcal {E}_{1}^{k}(\cdot )$ and $\mathcal {E}_{2}^{k}(\cdot )$ operators. By (5.11), the arguments of $\mathcal {E}_{1}^{k}$ and $\mathcal {E}_{2}^{k}$ in each term above equal

(5.30) $$ \begin{align} \frac{1}{2}\left(e^{tK_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{tK_{k}^{\oplus}}+e^{-tK_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-tK_{k}^{\oplus}}\right)-h_{k}^{\oplus} \end{align} $$

and

(5.31) $$ \begin{align} \frac{1}{2}\left(e^{tK_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}+B_{k}^{\oplus}\right)e^{tK_{k}^{\oplus}}-e^{-tK_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}-B_{k}^{\oplus}\right)e^{-tK_{k}^{\oplus}}\right), \end{align} $$

respectively. By the same identities that we used in the preceding proposition, it holds that

(5.32) $$ \begin{align} & e^{\pm tK_{k}^{\oplus}}\left(h_{k}^{\oplus}+A_{k}^{\oplus}\pm B_{k}^{\oplus}\right)e^{\pm tK_{k}^{\oplus}}\\ &\quad =\frac{1}{2}\left(\begin{array}{cc} e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}+e^{-tK_{k}}h_{k}e^{-tK_{k}} & \pm\left(e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}-e^{-tK_{k}}h_{k}e^{-tK_{k}}\right)\\ \pm\left(e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}-e^{-tK_{k}}h_{k}e^{-tK_{k}}\right) & e^{tK_{k}}\left(h_{k}+2P_{v_{k}}\right)e^{tK_{k}}+e^{-tK_{k}}h_{k}e^{-tK_{k}} \end{array}\right)\nonumber, \end{align} $$

and the claim follows.

6 Analysis of the exchange terms

In the preceding section, we accomplished a major qualitative goal of this paper, which was diagonalizing the bosonizable terms $H_{\operatorname {\mathrm {kin}}}^{\prime }+\sum _{k\in S_C}H_{\text {int}}^{k}$ in an explicit, quasi-bosonic fashion. In this section, we begin the quantitative study of the quasi-bosonic expression in Proposition 5.7.

The aim of this section is to estimate the $\mathcal {E}_{1}^{k}(\cdot )$ , $\mathcal {E}_{2}^{k}(\cdot )$ operators, which enter in the error terms due to the presence of the exchange correction $\varepsilon _{k,l}(\varphi ;\psi )$ in the quasi-bosonic commutation relations. We will therefore refer to them as exchange terms. Since these expressions are complicated, we thus devote three subsections to the analysis of them. In the first, we carry out a reduction procedure, in which we systematically consider the type of terms that can appear in the sums defining $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ for given $A,B$ , and reduce these to simpler expressions, or schematic forms. In doing so, we will see that every term appearing in $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ can for the purpose of estimation be sorted into one of four schematic forms. In the second subsection, we provide some basic commutator estimates associated with the four schematic forms, and in the final subsection we then carry out the quantitative analysis of these four forms to obtain the desired estimates of $\mathcal {E}_{1}^{k}(\cdot )$ and $\mathcal {E}_{2}^{k}(\cdot )$ .

6.1 Reduction to simpler expressions

Recall that for $k\in S_C$ and symmetric operators $A,B:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , we already defined $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ in Proposition 5.2. Since these expressions are complicated, it is helpful to discuss the general structure of $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ . Consider the first term of $\mathcal {E}_{1}^{k}(A)$ , which upon expansion is

(6.1) $$ \begin{align} & \left\{ b_{k}^{\ast}(Ae_{p}),\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}\left(e_{p},e_{q}\right)\right\} \right\} \nonumber \\ &\quad =b_{k}^{\ast}(Ae_{p})\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}\left(e_{p},e_{q}\right)\right\} +\left\{ b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right),\varepsilon_{k,l}\left(e_{p},e_{q}\right)\right\} b_{k}^{\ast}(Ae_{p})\\ &\quad =b_{k}^{\ast}(Ae_{p})b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\varepsilon_{k,l}\left(e_{p},e_{q}\right)+b_{k}^{\ast}(Ae_{p})\varepsilon_{k,l}\left(e_{p},e_{q}\right)b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\nonumber \\ &\qquad +b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)\varepsilon_{k,l}\left(e_{p},e_{q}\right)b_{k}^{\ast}(Ae_{p})+\varepsilon_{k,l}\left(e_{p},e_{q}\right)b_{l}^{\ast}\left(K_{l}^{\oplus}e_{q}\right)b_{k}^{\ast}(Ae_{p}),\nonumber \end{align} $$

which we may expand further using

(6.2) $$ \begin{align} \varepsilon_{k,l}\left(e_{p},e_{q}\right)=\varepsilon\left(\overline{k,p};\overline{l,q}\right)=-\left(\delta_{p,q}c_{\overline{q-l}}c_{\overline{p-k}}^{\ast}+\delta_{\overline{p-k},\overline{q-l}}c_{q}^{\ast}c_{p}\right) \end{align} $$

and then removing the delta on a case-by-case basis. This causes the sums over $p\in L_{k}^{\pm }$ and $q\in L_{l}^{\pm }$ of any of these terms to reduce to one of the schematic forms

(6.3) $$ \begin{align} &\sum_{p\in S}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}},\quad\sum_{p\in S}b_{k}^{\natural}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right),\quad \nonumber\\ & \quad \sum_{p\in S}\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right) \end{align} $$

subject to the following: S is a subset of $L_{k}^{\pm }\cap L_{l}^{\pm }$ , $b_{k}^{\natural }$ can denote either $b_{k}$ or $b_{k}^{\ast }$ , $\varepsilon _{k,l}(e_{p},e_{q})$ may instead be $\varepsilon _{l,k}(e_{q},e_{p})=\varepsilon _{k,l}(e_{p},e_{q})^{\ast }$ , T denotes either A or B, the terms $b_{k}^{\natural }(Te_{p})$ and $b_{l}^{\natural }(K_{l}^{\oplus }e_{q})$ may be interchanged, the notation

(6.4) $$ \begin{align} \tilde{c}_{p}=\begin{cases} c_{p} & p\in B_{F}^{c}\\ c_{p}^{\ast} & p\in B_{F} \end{cases} \end{align} $$

encodes the correct type of creation/annihilation operator depending on whether p corresponds to a hole state or an excited state, and $p_{1},p_{2},p_{3},p_{4}$ denote indices which depend on p.

The same decomposition holds for every term appearing in either $\mathcal {E}_{1}^{k}(A)$ or $\mathcal {E}_{2}^{k}(B)$ , so we must consider the forms of (6.3).

The only important feature of the dependency that the $p_{i}$ have with respect to p is that regardless of the term, when summing over $p\in S$ , $p_{i}$ ranges either exclusively over excited states (i.e., $p_{i}\in L_{k}^{\pm }$ ) or exclusively over hole states (i.e., $p_{i}\in \left(L_{k}-k\right)\cup \left(L_{-k}+k\right)$ or the analogous set for $L_{l}^{\pm }$ ), and that the assignments $p\mapsto p_{i}$ (for a given term) are injective. (Additionally, $p_{1}$ and $p_{2}$ will always be excited states.)

Therefore, when estimating, we can always expand the sum to either all of $B_{F}$ or all of $B_{F}^{c}$ , which is why the exact identities of S and the $p_{i}$ are of no importance to the estimation. For example,

(6.5) $$ \begin{align} \left|\sum_{p\in S}\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}\Psi\right\rangle \right| \leq\sum_{p\in S}\left\Vert \tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert \leq\sqrt{\sum_{p\in S}\left\Vert \tilde{c}_{p_{3}}\Psi\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert ^{2}} \le \left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle \end{align} $$

independently of S, $p_{3}$ and $p_{4}$ . Here, the two situations when both $p_{3}$ and $p_{4}$ range over excited states, and when both $p_{3}$ and $p_{4}$ range over hole states, can be treated similarly thanks to the particle-hole symmetry (1.13).

Discussion of estimation strategy

We conclude that both $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ reduce to sums over $l\in S_{C}$ of finitely many terms of the schematic forms of equation (6.3), so it suffices to estimate these. To this end, we must first perform some additional algebraic manipulation.

To motivate our goal, let us first derive a simple but insufficent estimate for one of these terms:

(6.6) $$ \begin{align} \sum_{p\in S}b_{k}^{\ast}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right). \end{align} $$

Using $\left\Vert c_{p}\right\Vert _{\text {Op}}=1$ , Proposition 4.4 and the Cauchy–Schwarz inequality, we find that

(6.7) $$ \begin{align} & \sum_{p\in S}\left|\left\langle \Psi,b_{k}^{\ast}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\Psi\right\rangle \right| \leq\sum_{p\in S}\left\Vert b_{k}\left(Te_{p_{1}}\right)\Psi\right\Vert \left\Vert b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\Psi\right\Vert \\ &\quad \leq\sum_{p\in S}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p_{1}}\right\Vert \left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}e_{p_{2}}\right\Vert \left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \leq\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}T\right\Vert _{\operatorname{\mathrm{HS}}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}\right\Vert _{\operatorname{\mathrm{HS}}}\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \nonumber \end{align} $$

for any $\Psi \in \mathcal {H}_{N}$ . To get a feeling for the quality of this estimate, we must know what to expect of the quantities on the right-hand side. We will see in the next sections that $\big\Vert (h_{l}^{\oplus })^{-\frac {1}{2}}K_{l}^{\oplus }\big\Vert _{\operatorname {\mathrm {HS}}} \leq O(k_{F}^{-\frac {1}{3} + \epsilon })$ . In general, what will take the place of T will be the $A_{k} (t )$ and $B_{k} (t )$ operators we defined in the last section, but as a simple example we consider

(6.8) $$ \begin{align} T= \left(\begin{array}{cc} P_{v_{k}} & 0\\ 0 & P_{v_{k}} \end{array}\right),\quad P_{v_{k}} = |v_k\rangle \langle v_k|, \quad v_{k}=\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sum_{p\in L_{k}}e_{p} \in \ell^{2}(L_{k}) \end{align} $$

for which

(6.9) $$ \begin{align} \left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}T\right\Vert _{\operatorname{\mathrm{HS}}} & =2\left\Vert h_{k}^{-\frac{1}{2}}P_{v_{k}}\right\Vert _{\operatorname{\mathrm{HS}}}=2\sqrt{\operatorname{\mathrm{tr}}\left(P_{v_{k}}h_{k}^{-1}P_{v_{k}}\right)}\\ & =2\left\Vert v_{k}\right\Vert \sqrt{\left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle }=\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|^{\frac 1 2} \sqrt{\sum_{p\in L_{k}}\lambda_{k,p}^{-1}}\leq O\big(k_{F}^{\frac{1}{2}} \big)\nonumber \end{align} $$

when $|k|\sim 1$ . Here, we used $|L_k|\le C|k| k_F^2$ and the bound $\sum _{p\in L_{k}}\lambda _{k,p}^{-1}\leq Ck_{F}$ from Proposition A.2. Thus, for any state satisfying $\left\langle \Psi ,H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi \right\rangle \le O(k_F)$ (c.f. Theorem 1.2), the overall estimate for the right side of (6.7) is $O(k_{F}^{\frac {7}{6} + \epsilon })$ which is insufficient as the correlation energy is of order $k_F$ .

The technical issue with the estimation in (6.7) lies in only using that $\left\Vert c_{p}\right\Vert _{\text {Op}}=1$ , for we may get better bounds by using $\left\langle \Psi ,\mathcal {N}_{E} H^{\prime }_{\mathrm {kin}}\Psi \right\rangle $ instead of $\left\langle \Psi ,H^{\prime }_{\mathrm {kin}}\Psi \right\rangle $ . For example,

(6.10) $$ \begin{align} &\quad\,\sum_{p\in S}\left|\left\langle \Psi,b_{k}^{\ast}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{l} \left(K_{l}^{\oplus}e_{p_{2}}\right)\Psi\right\rangle \right| \end{align} $$

(6.11) $$ \begin{align} &=\sum_{p\in S}\left|\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}b_{k}^{\ast}\left(Te_{p_{1}}\right)b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\rangle \right|\leq\sum_{p\in S}\left\Vert b_{k}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & \leq\sum_{p\in S}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p_{1}}\right\Vert \left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{3}}\Psi,H_{\operatorname{\mathrm{kin}}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{3}}\Psi\right\rangle \left\langle \tilde{c}_{p_{4}}\Psi,H_{\operatorname{\mathrm{kin}}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{4}}\Psi\right\rangle } \end{align} $$

$$\begin{align*} & \leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert \right)\sqrt{\sum_{p\in S}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}e_{q}\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}H_{\operatorname{\mathrm{kin}}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{3}}\Psi\right\rangle }\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\nonumber \\ & \leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert \right)\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}\right\Vert _{\operatorname{\mathrm{HS}}}\sqrt{\left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle },\end{align*}$$

where we used that $\left[\tilde {c}_{p},b_{k}(\cdot )\right]=0$ (as we will see in Proposition 6.1 below) and momentarily looked ahead to the definition (6.30) for $H_{\operatorname {\mathrm {kin}}}^{\prime \left(\pm 1\right)}$ and Lemma 6.6 (we take supremum over $p_4$ and sum over $p_3$ to get the second inequality). Considering again the example in (6.8), we find

(6.12) $$ \begin{align} \max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert =\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\left\Vert h_{k}^{-\frac{1}{2}}v_{k}\right\Vert =\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\sqrt{\sum_{k\in L_{k}}\lambda_{k,p}^{-1}}\leq O(k_{F}^{-\frac{1}{2}}). \end{align} $$

Thus, for any state satisfying $\left\langle \Psi ,H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi \right\rangle \le O(k_F)$ and $\left\langle \Psi ,\mathcal {N}_{E}H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi \right\rangle \le O(k_F^2)$ (c.f. Theorem 1.2), the right side of (6.10) is thus bounded by $O(k_{F}^{ \frac {2}{3} + \epsilon })$ , which is much smaller than the correlation energy.

Our goal is, therefore, to reduce the schematic forms of equation (6.3) to those of the form $\sum _{p\in S}\tilde {c}_{p_{3}}^{\ast }b_{k}^{\natural }(Te_{p_{1}})b_{l}^{\natural }(K_{l}^{\oplus }e_{p_{2}})\tilde {c}_{p_{4}}$ , which we may then estimate as above. While $\left[\tilde {c}_{p},b_{k}(\cdot )\right]=0$ , it is generally the case that $\left[\tilde {c}_{p},b_{k}^{\ast }(\cdot )\right]\neq 0$ , so this will also introduce additional commutator terms which we must then estimate separately.

Taking into account whether $b_{k}^{\natural }=b_{k}$ or $b_{k}^{\natural }=b_{k}^{\ast }$ , the schematic forms of equation (6.3) are either of the form (supressing the summation, the arguments and the subscripts for brevity)

(6.13) $$ \begin{align} b^{\ast}b^{\ast}\tilde{c}^{\ast}\tilde{c},\quad b^{\ast}b\tilde{c}^{\ast}\tilde{c},\quad bb^{\ast}\tilde{c}^{\ast}\tilde{c},\quad bb\tilde{c}^{\ast}\tilde{c},\quad b^{\ast}\tilde{c}^{\ast}\tilde{c}b,\quad b\tilde{c}^{\ast}\tilde{c}b,\quad b\tilde{c}^{\ast}\tilde{c}b^{\ast}, \end{align} $$

or reduce to one of these by taking the adjoint, which as we will estimate, $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ as bilinear forms does not matter. Using that commutators of the form $\left[b,\tilde {c}\right]$ , $\left[b^{\ast },\tilde {c}^{\ast }\right]$ and $\left[b,\left[b,\tilde {c}^{\ast }\right]\right]$ vanish (verified below), these schematic forms reduce to

(6.14) $$ \begin{align} b^{\ast}b^{\ast}\tilde{c}^{\ast}\tilde{c} & =\tilde{c}^{\ast}b^{\ast}b^{\ast}\tilde{c} ,\nonumber \\ b^{\ast}b\tilde{c}^{\ast}\tilde{c} & =\tilde{c}^{\ast}b^{\ast}b\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b^{\ast}\tilde{c}+\left[b^{\ast},\left[b,\tilde{c}^{\ast}\right]\right]\tilde{c},\nonumber \\ bb^{\ast}\tilde{c}^{\ast}\tilde{c} & =\tilde{c}^{\ast}bb^{\ast}\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b^{\ast}\tilde{c},\nonumber \\ bb\tilde{c}^{\ast}\tilde{c} & =\tilde{c}^{\ast}bb\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b\tilde{c},\\ b^{\ast}\tilde{c}^{\ast}\tilde{c}b & =\tilde{c}^{\ast}b^{\ast}b\tilde{c},\nonumber \\ b\tilde{c}^{\ast}\tilde{c}b & =\tilde{c}^{\ast}bb\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b\tilde{c},\nonumber \\ b\tilde{c}^{\ast}\tilde{c}b^{\ast} & =\tilde{c}^{\ast}bb^{\ast}\tilde{c}+\left[b,\tilde{c}^{\ast}\right]b^{\ast}\tilde{c}+\tilde{c}^{\ast}b\left[\tilde{c},b^{\ast}\right]+\left[b,\tilde{c}^{\ast}\right]\left[\tilde{c},b^{\ast}\right].\nonumber \end{align} $$

Reintroducing the $b^{\natural }$ notation outside the commutators and using once more our freedom to take adjoints, we find that every term on the right-hand sides of the two equations above takes one of the four schematic forms

(6.15) $$ \begin{align} \tilde{c}^{\ast}b^{\natural}b^{\natural}\tilde{c},\quad\left[\tilde{c},b^{\ast}\right]^{\ast}b^{\natural}\tilde{c},\quad\left[\left[\tilde{c},b^{\ast}\right],b\right]^{\ast}\tilde{c},\quad\left[\tilde{c},b^{\ast}\right]^{\ast}\left[\tilde{c},b^{\ast}\right]. \end{align} $$

These are the final forms which we will explicitly estimate.

6.2 Preliminary commutator estimates

In addition to the general estimates which we derived at the start of this section, we will also need estimates on the commutator terms which appear in the schematic forms of equation (6.23), which we now derive. First, we must, however, verify that the commutators $\left[b,\tilde {c}\right]$ , $\left[b^{\ast },\tilde {c}^{\ast }\right]$ and $\left[b,\left[b,\tilde {c}^{\ast }\right]\right]$ vanish, which we relied upon in our reduction procedure:

Proposition 6.1. For all $k,l\in \mathbb {Z}_{+}^{3}$ , $\varphi \in \ell ^{2}(L_{k}^{\pm })$ , $\psi \in \ell ^{2}\left(L_{l}^{\pm }\right)$ and $p\in \mathbb {Z}^{3}$ it holds that

$$ \begin{align*} \left[b_{k}(\varphi ),\tilde{c}_{p}\right]=\left[b_{k}^{\ast}(\varphi ),\tilde{c}_{p}^{\ast}\right] =0,\quad \left[b_{l}\left(\psi\right),\left[b_{k}(\varphi ),\tilde{c}_{p}^{\ast}\right]\right] =0. \end{align*} $$

Proof. We compute from the definitions that for any $q\in L_{k}^{\pm }$ ,

(6.16) $$ \begin{align} \left[b_{\overline{k,q}},\tilde{c}_{p}\right] & =\left[c_{\overline{q-k}}^{\ast}c_{q},\tilde{c}_{p}\right]=c_{\overline{q-k}}^{\ast}\left\{ c_{q},\tilde{c}_{p}\right\} -\left\{ c_{\overline{q-k}}^{\ast},\tilde{c}_{p}\right\} c_{q}\nonumber \\ & =\begin{cases} c_{\overline{q-k}}^{\ast}\left\{ c_{q},c_{p}\right\} -\left\{ c_{\overline{q-k}}^{\ast},c_{p}\right\} c_{q} ,& p\in B_{F}^{c}\nonumber\\ c_{\overline{q-k}}^{\ast}\left\{ c_{q},c_{p}^{\ast}\right\} -\left\{ c_{\overline{q-k}}^{\ast},c_{p}^{\ast}\right\} c_{q} ,& p\in B_{F} \end{cases}\\ & =0 \end{align} $$

as all anticommutators on the second line vanish either directly by the CAR or by disjointness of $B_{F}$ and $B_{F}^{c}$ . By linearity, $\left[b_{k}(\varphi ),\tilde {c}_{p}\right]=0$ , and $\left[b_{k}^{\ast }(\varphi ),\tilde {c}_{p}^{\ast }\right]=-\left[b_{k}(\varphi ),\tilde {c}_{p}\right]^{\ast }=0$ .

For the double commutator, we first compute $[b_{\overline {k,q}},\tilde {c}_{p}^{\ast }]$ . As above, we find

$$\begin{align*}\left[b_{\overline{k,q}},\tilde{c}_{p}^{\ast}\right] =\begin{cases} c_{\overline{q-k}}^{\ast}\left\{ c_{q},c_{p}^{\ast}\right\} -\left\{ c_{\overline{q-k}}^{\ast},c_{p}^{\ast}\right\} c_{q} ,& p\in B_{F}^{c}\\ c_{\overline{q-k}}^{\ast}\left\{ c_{q},c_{p}\right\} -\left\{ c_{\overline{q-k}}^{\ast},c_{p}\right\} c_{q}, & p\in B_{F} \end{cases} =\begin{cases} \delta_{q,p}\tilde{c}_{\overline{q-k}} ,& p\in B_{F}^{c}\\ -\delta_{\overline{q-k},p}\tilde{c}_{q}, & p\in B_{F} \end{cases}\end{align*}$$

so

(6.17) $$ \begin{align} \left[b_{k}(\varphi ),\tilde{c}_{p}^{\ast}\right] & =\sum_{q\in L_{k}^{\pm}}\left\langle \varphi,e_{q}\right\rangle \left[b_{\overline{k,q}},\tilde{c}_{p}^{\ast}\right]=\sum_{q\in L_{k}^{\pm}}\left\langle \varphi,e_{q}\right\rangle \begin{cases} \delta_{q,p}\tilde{c}_{\overline{q-k}} & p\in B_{F}^{c} \nonumber\\ -\delta_{\overline{q-k},p}\tilde{c}_{q} & p\in B_{F} \end{cases}\\ & =1_{L_{k}^{\pm}}(p)\left\langle \varphi,e_{p}\right\rangle \tilde{c}_{\overline{p-k}}-1_{L_{k}-k}(p)\left\langle \varphi,e_{p+k}\right\rangle \tilde{c}_{p+k}-1_{L_{k}+k}(p)\left\langle \varphi,e_{p-k}\right\rangle \tilde{c}_{p-k}, \end{align} $$

where $1_{S}(\cdot )$ denotes the indicator function of a set S. Observing that $\left[b_{k}(\varphi ),\tilde {c}_{p}^{\ast }\right]$ is a linear combination of $\tilde {c}_{p}$ terms, we conclude that $\left[b_{l}(\psi ),\left[b_{k}(\varphi ),\tilde {c}_{p}^{\ast }\right]\right]=0$ by the first part.

We now move into the estimation of the nonvanishing commutators. We begin with the single commutator; we state the estimate and make a remark:

Proposition 6.2. For all $k\in \mathbb {Z}_{+}^{3}$ , sequences $\left(\varphi _{p}\right)_{p\in \mathbb {Z}^{3}}\in \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$ \begin{align*} \sum_{p\in\mathbb{Z}^{3}}\left\Vert \left[\tilde{c}_{p},b_{k}^{\ast}\left(\varphi_{p}\right)\right]\Psi\right\Vert ^{2} & \leq3\left(\sum_{p\in L_{k}^{\pm}}\max_{q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left\Vert \Psi\right\Vert ^{2}\\ \sum_{p\in\mathbb{Z}^{3}}\left\Vert \left[\tilde{c}_{p}^{\ast},b_{k}\left(\varphi_{p}\right)\right]\Psi\right\Vert ^{2} & \leq4\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align*} $$

Remark 6.1. The statement may appear overly general in that it involves general sequences $(\varphi_{p})_{p\in \mathbb {Z}^{3}}\subset \ell ^{2}(L_{k}^{\pm })$ rather than the explicit vectors $(Te_{p_{1}})_{p\in S}\subset \ell ^{2}(L_{k}^{\pm })$ that we must consider. The point of the generality is, however, only to avoid having to explicitly state the dependencies of the set S and the $p_{i}$ ’s of each possible schematic form, as independently of these it is easy to see that a sum such as $\sum _{p\in S}\left\Vert [\tilde {c}_{p_{3}},b_{k}^{\ast }(Te_{p_{1}})]\Psi \right\Vert ^{2}$ can always be cast into the form in the statement.

Proof. Taking the adjoint of equation (6.17) yields

(6.18) $$ \begin{align} \left[\tilde{c}_{p},b_{k}^{\ast}(\varphi )\right]=1_{L_{k}^{\pm}}(p)\left\langle e_{p},\varphi\right\rangle \tilde{c}_{\overline{p-k}}^{\ast}-1_{L_{k}-k}(p)\left\langle e_{p+k},\varphi\right\rangle \tilde{c}_{p+k}^{\ast}-1_{L_{k}+k}(p)\left\langle e_{p-k},\varphi\right\rangle \tilde{c}_{p-k}^{\ast}, \end{align} $$

and so we can for any $\Psi \in \mathcal {H}_{N}$ estimate by the (squared) triangle inequality, using also that $L_{k}^{\pm }$ and $\left(L_{k}-k\right)\cap \left(L_{-k}+k\right)$ are disjoint and $\left\Vert \tilde {c}_{p}^{\ast }\right\Vert _{\text {Op}}=1$ , that

(6.19) $$ \begin{align} &\sum_{p\in\mathbb{Z}^{3}}\left\Vert \left[\tilde{c}_{p},b_{k}^{\ast}\left(\varphi_{p}\right)\right]\Psi\right\Vert ^{2} \nonumber \\ &\leq\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{\overline{p-k}}^{\ast}\Psi\right\Vert ^{2} +2\sum_{p\in L_{k}-k}\left|\left\langle e_{p+k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{p+k}^{\ast}\Psi\right\Vert ^{2} +2\sum_{p\in L_{-k}+k}\left|\left\langle e_{p-k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{p-k}^{\ast}\Psi\right\Vert ^{2}\nonumber \\ & \leq\left(\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi_{p}\right\rangle \right|^{2}+2\sum_{p\in L_{k}-k}\left|\left\langle e_{p+k},\varphi_{p}\right\rangle \right|^{2}+2\sum_{p\in L_{-k}+k}\left|\left\langle e_{p-k},\varphi_{p}\right\rangle \right|^{2}\right)\left\Vert \Psi\right\Vert ^{2} \nonumber\\ & \leq\left(\sum_{p\in L_{k}^{\pm}}\max_{q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}+2\sum_{p\in L_{k}}\max_{q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}+2\sum_{p\in L_{-k}}\max_{q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left\Vert \Psi\right\Vert ^{2}\nonumber \\ & =3\left(\sum_{p\in L_{k}^{\pm}}\max_{q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left\Vert \Psi\right\Vert ^{2}, \end{align} $$

which implies the first estimate. For the second estimate, we find in a similar manner (now directly from equation (6.17)) that

(6.20) $$ \begin{align} &\sum_{p\in\mathbb{Z}^{3}}\left\Vert \left[\tilde{c}_{p}^{\ast},b_{k}\left(\varphi_{p}\right)\right]\Psi\right\Vert ^{2} \leq\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{\overline{p-k}}\Psi\right\Vert ^{2} +2\sum_{p\in L_{k}-k}\left|\left\langle e_{p+k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{p+k}\Psi\right\Vert ^{2} \nonumber \\ &\qquad +2\sum_{p\in L_{-k}+k}\left|\left\langle e_{p-k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \tilde{c}_{p-k}\Psi\right\Vert ^{2} \nonumber\\ &\quad \leq\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left(\sum_{p\in L_{k}^{\pm}}\left\Vert \tilde{c}_{\overline{p-k}}\Psi\right\Vert ^{2}+2\sum_{p\in L_{k}-k}\left\Vert \tilde{c}_{p+k}\Psi\right\Vert ^{2}+2\sum_{p\in L_{-k}+k}\left\Vert \tilde{c}_{p-k}\Psi\right\Vert ^{2}\right)\nonumber \\ &\quad =\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left(\sum_{p\in L_{k}^{\pm}}\left\Vert c_{\overline{p-k}}^{\ast}\Psi\right\Vert ^{2}+2\sum_{p\in L_{k}}\left\Vert c_{p}\Psi\right\Vert ^{2}+2\sum_{p\in L_{-k}}\left\Vert c_{p}\Psi\right\Vert ^{2}\right)\nonumber \\ &\quad \leq4\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align} $$

Lastly, we estimate the double commutator:

Proposition 6.3. For all $k,l\in \mathbb {Z}_{+}^{3}$ , sequences $\left(\varphi _{p}\right)_{p\in \mathbb {Z}^{3}}\subset \ell ^{2}(L_{k}^{\pm })$ and $\left(\psi _{p}\right)_{p\in \mathbb {Z}^{3}}\subset \ell ^{2}\left(L_{l}^{\pm }\right)$ , and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\sum_{p\in\mathbb{Z}^{3}}\left\Vert \left[\left[\tilde{c}_{p},b_{k}^{\ast}\left(\varphi_{p}\right)\right],b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}\leq12\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\psi_{q}\right\rangle \right|^{2}\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align*}$$

Proof. From (6.18), we have that

(6.21) $$ \begin{align} \left[\left[\tilde{c}_{p},b_{k}^{\ast}\left(\varphi_{p}\right)\right],b_{l}\left(\psi_{p}\right)\right] & =1_{L_{k}^{\pm}}(p)\left\langle e_{p},\varphi\right\rangle \left[\tilde{c}_{\overline{p-k}}^{\ast},b_{l}\left(\psi_{p}\right)\right]-1_{L_{k}-k}(p)\left\langle e_{p+k},\varphi\right\rangle \left[\tilde{c}_{p+k}^{\ast},b_{l}\left(\psi_{p}\right)\right]\\ &\quad -1_{L_{k}+k}(p)\left\langle e_{p-k},\varphi\right\rangle \left[\tilde{c}_{p-k}^{\ast},b_{l}\left(\psi_{p}\right)\right]\nonumber, \end{align} $$

and so, by the triangle inequality and the second estimate of Proposition 6.2,

(6.22) $$ \begin{align} & \sum_{p\in\mathbb{Z}^{n}}\left\Vert \left[\left[\tilde{c}_{p},b_{k}^{\ast}\left(\varphi_{p}\right)\right],b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}\leq\sum_{p\in L_{k}^{\pm}}\left|\left\langle e_{p},\varphi_{p}\right\rangle \right|^{2}\left\Vert \left[\tilde{c}_{\overline{p-k}}^{\ast},b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}\nonumber \\ &\qquad +2\sum_{p\in L_{k}-k}\left|\left\langle e_{p+k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \left[\tilde{c}_{p+k}^{\ast},b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}+2\sum_{p\in L_{-k}+k}\left|\left\langle e_{p-k},\varphi_{p}\right\rangle \right|^{2}\left\Vert \left[\tilde{c}_{p-k}^{\ast},b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}\nonumber \\ &\quad \leq\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left(\sum_{p\in L_{k}^{\pm}}\left\Vert \left[\tilde{c}_{\overline{p-k}}^{\ast},b_{l}\left(\psi_{p}\right)\right]\Psi\right\Vert ^{2}+2\sum_{p\in L_{k}}\left\Vert \left[\tilde{c}_{p}^{\ast},b_{l}\left(\psi_{p-k}\right)\right]\Psi\right\Vert ^{2}\right.\nonumber \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+2\sum_{p\in L_{-k}}\left\Vert \left[\tilde{c}_{p}^{\ast},b_{l}\left(\psi_{p+k}\right)\right]\Psi\right\Vert ^{2}\right) \nonumber\\ &\quad \leq12\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\varphi_{q}\right\rangle \right|^{2}\right)\left(\max_{p\in L_{k}^{\pm},q\in\mathbb{Z}^{3}}\left|\left\langle e_{p},\psi_{q}\right\rangle \right|^{2}\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \end{align} $$

6.3 Final estimation of the exchange terms

Now we are ready to derive bounds for the exchange terms $\mathcal {E}_{1}^{k}(A)$ and $\mathcal {E}_{2}^{k}(B)$ defined in Proposition 5.2. Recall that we have reduced the estimation of these complicated operators to the task of obtaining a uniform estimate for the four explicit forms

(6.23) $$ \begin{align} & \sum_{l\in S_{C}}\sum_{p\in S}\tilde{c}_{p_{3}}^{\ast}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}},\qquad\quad\;\;\sum_{l\in S_{C}}\sum_{p\in S}\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]^{\ast}b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}} \nonumber\\ & \sum_{l\in S_{C}}\sum_{p\in S}\left[\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right],b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]^{\ast}\tilde{c}_{p_{4}},\quad\sum_{l\in S_{C}}\sum_{p\in S}\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]^{\ast}\left[\tilde{c}_{p_{4}},b_{l}^{\ast}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right], \end{align} $$

subject to the following rules: $b_{k}^{\natural }$ denotes either $b_{k}$ or $b_{k}^{\ast }$ , T denotes either A or B, and $b_{k}^{\natural }\left(Te_{p_{1}}\right)$ and $b_{l}^{\natural }(K_{l}^{\oplus }e_{p_{2}})$ may be interchanged. Furthermore, the notation $\tilde {c}_{p}$ denotes either $c_{p}$ or $c_{p}^{\ast }$ as appropriate for p, and the set S is such that the assignments $p\mapsto p_{1},p_{2},p_{3},p_{4}$ are injective and map exclusively into $B_{F}$ or $B_{F}^{c}$ .

Let us start by giving estimates in terms of $\mathcal {N}_E^2$ . For the statement, we define the $\left\Vert \cdot \right\Vert _{\infty ,2}$ -norm of an operator $T:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ by

(6.24) $$ \begin{align} \left\Vert T\right\Vert _{\infty,2}=\sqrt{\sum_{p\in L_{k}^{\pm}}\max_{q\in L_{k}^{\pm}}\left|\left\langle e_{p},Te_{q}\right\rangle \right|^{2}}. \end{align} $$

This is a minor but necessary detail, as unlike the simple estimate of equation (6.10), we cannot take the maximum outside the sum for all schematic terms, so we need this slightly stronger norm. Note that

(6.25) $$ \begin{align} \max_{p,q\in L_{k}^{\pm}}\left|\left\langle e_{p},Te_{q}\right\rangle \right|\leq\max_{p\in L_{k}^{\pm}}\left\Vert Te_{p}\right\Vert \leq\left\Vert T\right\Vert _{\infty,2}. \end{align} $$

Now the estimate the follows.

Proposition 6.4. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $T:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in \mathcal {H}_{N}$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,\mathcal{E}_{i}^{k}(T)\Psi\right\rangle \right|\leq C\left\Vert T\right\Vert _{\infty,2}\left(\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right)\left\langle \Psi,\left(1+\mathcal{N}_{E}^{2}\right)\Psi\right\rangle \end{align*}$$

with $i=1,2$ , for a constant $C>0$ independent of all relevant quantities.

Proof. We estimate each schematic form of (6.23) using the estimates of the Propositions 4.2, 6.2, 6.3 and Lemma 4.3, as well as the Cauchy-Schwarz inequality. First is $\tilde {c}^{\ast }b^{\natural }b^{\natural }\tilde {c}$ :

(6.26) $$ \begin{align} & \qquad\,\sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\rangle \right|\leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert b_{k}^\sharp \left(Te_{p_{1}}\right)^\ast\tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & \leq C\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert Te_{p_{1}}\right\Vert \left\Vert K_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{3}}\Psi,\left(1+\mathcal{N}_{E}^{\left(\pm1\right)}\right)\tilde{c}_{p_{3}}\Psi\right\rangle \left\langle \tilde{c}_{p_{4}}\Psi,\left(1+\mathcal{N}_{E}^{\left(\pm1\right)}\right)\tilde{c}_{p_{4}}\Psi\right\rangle }\\ & \leq C\max_{p\in L_{k}^{\pm}}\left\Vert Te_{p}\right\Vert \sum_{l\in S_{C}}\max_{q\in L_{l}^{\pm}}\left\Vert K_{l}^{\oplus}e_{q}\right\Vert \sqrt{\sum_{p\in S}\left\langle \Psi,\left(\tilde{c}_{p_{3}}^{\ast}\mathcal{N}_{E}^{\left(\pm1\right)}\tilde{c}_{p_{3}}+\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{3}}\right)\Psi\right\rangle }\nonumber \\ & \quad\;\cdot\sqrt{\sum_{p\in S}\left\langle \Psi,\left(\tilde{c}_{p_{4}}^{\ast}\mathcal{N}_{E}^{\left(\pm1\right)}\tilde{c}_{p_{4}}+\tilde{c}_{p_{4}}^{\ast}\tilde{c}_{p_{4}}\right)\Psi\right\rangle }\leq C\left\Vert T\right\Vert _{\infty,2}\left(\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right)\left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle. \nonumber \end{align} $$

Then, $\left[\tilde {c},b^{\ast }\right]^{\ast }b^{\natural }\tilde {c}$ :

(6.27) $$ \begin{align} & \quad\,\sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]^{\ast}b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\rangle \right| \nonumber\\ &\leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]\Psi\right\Vert \left\Vert b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & \leq C\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]\Psi\right\Vert \left\Vert K_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{4}}\Psi,\left(1+\mathcal{N}_{E}^{\left(\pm1\right)}\right)\tilde{c}_{p_{4}}\Psi\right\rangle }\nonumber \\ & \leq C\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\sqrt{\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]\Psi\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\langle \tilde{c}_{p_{4}}\Psi,\left(1+\mathcal{N}_{E}^{\left(\pm1\right)}\right)\tilde{c}_{p_{4}}\Psi\right\rangle }\\ & \leq C\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\sqrt{\sum_{p\in L_{k}^{\pm}}\max_{q\in L_{k}^{\pm}}\left|\left\langle e_{p},Te_{q}\right\rangle \right|^{2}\left\Vert \Psi\right\Vert ^{2}}\sqrt{\left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle }\nonumber \\ & \leq C\left\Vert T\right\Vert _{\infty,2}\left(\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right)\left\Vert \Psi\right\Vert \sqrt{\left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle }.\nonumber \end{align} $$

Now, $\left[\left[\tilde {c},b^{\ast }\right],b\right]^{\ast }\tilde {c}$ :

(6.28) $$ \begin{align} & \quad\,\sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\left[\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right],b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]^{\ast}\tilde{c}_{p_{4}}\Psi\right\rangle \right|\nonumber \\ & \leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left[\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right],b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]\Psi\right\Vert \left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & \leq\sum_{l\in S_{C}}\sqrt{\sum_{p\in S}\left\Vert \left[\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right],b_{l}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]\Psi\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert ^{2}}\\ & \leq C\sum_{l\in S_{C}}\sqrt{\left(\max_{p,q\in L_{k}^{\pm}}\left|\left\langle e_{p},Te_{q}\right\rangle \right|^{2}\right)\left(\max_{p,q\in L_{l}^{\pm}}\left|\left\langle e_{p},K_{l}^{\oplus}e_{q}\right\rangle \right|^{2}\right)\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle }\sqrt{\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle }\nonumber \\ & \leq C\left\Vert T\right\Vert _{\infty,2}\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\left\langle \Psi,\mathcal{N}_{E}\Psi\right\rangle. \nonumber \end{align} $$

And finally, $\left[\tilde {c},b^{\ast }\right]^{\ast }\left[\tilde {c},b^{\ast }\right]$ :

(6.29) $$ \begin{align} & \quad\,\sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]^{\ast}\left[\tilde{c}_{p_{4}},b_{l}^{\ast}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]\Psi\right\rangle \right|\nonumber \\ & \leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]\Psi\right\Vert \left\Vert \left[\tilde{c}_{p_{4}},b_{l}^{\ast}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]\Psi\right\Vert \nonumber \\ & \leq\sum_{l\in S_{C}}\sqrt{\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{3}},b_{k}^{\ast}\left(Te_{p_{1}}\right)\right]\Psi\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\Vert \left[\tilde{c}_{p_{4}},b_{l}^{\ast}\left(K_{l}^{\oplus}e_{p_{2}}\right)\right]\Psi\right\Vert ^{2}}\\ & \leq C\sum_{l\in S_{C}}\sqrt{\sum_{p\in L_{k}^{\pm}}\max_{q\in L_{k}^{\pm}}\left|\left\langle e_{p},Te_{q}\right\rangle \right|^{2}\left\Vert \Psi\right\Vert ^{2}}\sqrt{\sum_{p\in L_{l}^{\pm}}\max_{q\in L_{l}^{\pm}}\left|\left\langle e_{p},K_{l}^{\oplus}e_{q}\right\rangle \right|^{2}\left\Vert \Psi\right\Vert ^{2}}\nonumber \\ & \leq C\left\Vert T\right\Vert _{\infty,2}\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\left\Vert \Psi\right\Vert ^{2}.\nonumber \end{align} $$

Now we derive a kinetic bound.

Proposition 6.5. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $T:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ ,

$$ \begin{align*} \left|\left\langle \Psi,\mathcal{E}_{i}^{k}(T)\Psi\right\rangle \right| &\le C \sum_{l\in S_{C}}\left(\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}\right\Vert _{\operatorname{\mathrm{HS}}}+\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right) \\ &\quad \times \Bigg[ \left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert \right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\\ &\quad +\left\Vert T\right\Vert _{\infty,2} \left(\left\langle \Psi,\left(1+H_{\operatorname{\mathrm{kin}}}^{\prime}\right)\Psi\right\rangle +\left\Vert \Psi\right\Vert \sqrt{\left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\right) \Bigg] \end{align*} $$

for $i=1,2$ , for a constant $C>0$ independent of all relevant quantities.

As a technical preparation, let us observe that from (1.14) we may associate to $H_{\mathrm {kin}}'$ the operators

(6.30) $$ \begin{align} H_{\text{kin}}^{\prime\left(\pm1\right)}=\sum_{p\in B_{F}^{c}}\vert|p|^{2}-\zeta\vert\,c_{p}^{\ast}c_{p}+\sum_{p\in B_{F}}\vert|p|^{2}-\zeta\vert\,c_{p}c_{p}^{\ast} \end{align} $$

acting on $\mathcal {H}_{N\pm 1}$ (the expressions of $H_{\text {kin}}^{\prime \left(+1\right)}$ and $H_{\text {kin}}^{\prime \left(-1\right)}$ are the same, but the domains are different). With this interpretation, we have the following lemma (c.f. Lemma 4.3):

Lemma 6.6. It holds that

$$ \begin{align*}\tilde{c}_{p}^{\ast}H_{\operatorname{\mathrm{kin}}}^{\prime\left(\pm1\right)}\tilde{c}_{p}\leq H_{\operatorname{\mathrm{kin}}}^{\prime}\end{align*} $$

for all $p\in \mathbb {Z}^{3}$ and

$$\begin{align*}\sum_{p\in B_{F}^{c}}c_{p}^{\ast}H_{\operatorname{\mathrm{kin}}}^{\prime\left(-1\right)}c_{p} \le \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}, \quad \sum_{p\in B_{F}}c_{p}H_{\operatorname{\mathrm{kin}}}^{\prime\left(+1\right)}c_{p}^{\ast}\leq\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align*}$$

Proof. By the CAR, we have that

(6.31) $$ \begin{align} & \quad\quad\sum_{p\in B_{F}^{c}}c_{p}^{\ast}H_{\text{kin}}^{\prime\left(-1\right)}c_{p}=\sum_{p\in B_{F}^{c}}c_{p}^{\ast}\left(\sum_{q\in B_{F}^{c}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,c_{q}^{\ast}c_{q}+\sum_{q\in B_{F}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,c_{q}c_{q}^{\ast}\right)c_{p}\nonumber\\ & =\left(\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}\right)\left(\sum_{q\in B_{F}^{c}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,c_{q}^{\ast}c_{q}+\sum_{q\in B_{F}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,c_{q}c_{q}^{\ast}\right)+\sum_{p\in B_{F}^{c}}c_{p}^{\ast}\left[\sum_{q\in B_{F}^{c}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,c_{q}^{\ast}c_{q},c_{p}\right]\nonumber \\ & =\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}-\sum_{p,q\in B_{F}^{c}}\vert\left|q\right|^{2}-\zeta_{0}\vert\,\delta_{p,q}c_{p}^{\ast}c_{q} \leq\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}, \end{align} $$

and the inequality for $c_{p}H_{\text {kin}}^{\prime \left(+1\right)}c_{p}^{\ast }$ can be derived similarly. That $\tilde {c}_{p}^{\ast }H_{\text {kin}}^{\prime \left(\pm 1\right)}\tilde {c}_{p}\leq H_{\operatorname {\mathrm {kin}}}^{\prime }$ follows exactly as the inequality $\tilde {c}_{p}^{\ast }\mathcal {N}_{E}^{\left(\pm 1\right)}\tilde {c}_{p}\leq \mathcal {N}_{E}$ did in Lemma 4.3.

Now we are ready to give the

Proof of Proposition 6.5.

For all schematic forms except

(6.32) $$ \begin{align} \sum_{l\in S_{C}}\sum_{p\in S}\tilde{c}_{p_{3}}^{\ast}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}, \end{align} $$

we can use the estimates derived in Proposition 6.4, specifically the equations (6.27) through (6.29), and the fact that $\mathcal {N}_{E}\leq H_{\operatorname {\mathrm {kin}}}^{\prime }$ . For the schematic form in (6.32), we can by Proposition 4.4 estimate that

(6.33) $$ \begin{align} & \quad\,\sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}b_{k}^{\natural}\left(Te_{p_{1}}\right)b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\rangle \right|\leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert b_{k}^{\natural}\left(Te_{p_{1}}\right)\tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert b_{l}^{\natural}\left(K_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & \leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p_{1}}\right\Vert \left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{3}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{3}}\Psi\right\rangle \left\langle \tilde{c}_{p_{4}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{4}}\Psi\right\rangle }\nonumber \\ & +\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p_{1}}\right\Vert \left\Vert K_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{3}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{3}}\Psi\right\rangle }\left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert \nonumber\\ & +\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert Te_{p_{1}}\right\Vert \left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}e_{p_{2}}\right\Vert \left\Vert \tilde{c}_{p_{3}}\Psi\right\Vert \sqrt{\left\langle \tilde{c}_{p_{4}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{4}}\Psi\right\rangle }\nonumber \\ & +\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert Te_{p_{1}}\right\Vert \left\Vert K_{l}^{\oplus}e_{p_{2}}\right\Vert \left\Vert \tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert \tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ & =:A_{1}+A_{2}+A_{3}+A_{4}. \end{align} $$

The terms $A_{1}$ through $A_{4}$ can be estimated by the Cauchy-Schwarz inequality, Lemma 6.6, the inequality $\mathcal {N}_{E}\leq H_{\operatorname {\mathrm {kin}}}^{\prime }$ and the fact that $\max _{p\in L_{k}^{\pm }}\left\Vert Te_{p}\right\Vert \leq \left\Vert T\right\Vert _{\infty ,2}$ as

(6.34) $$ \begin{align} A_{1} & \leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert \right)\left(\sum_{l\in S_{C}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}\right\Vert _{\text{HS}}\right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle },\nonumber \\ A_{2} & \leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Te_{p}\right\Vert \right)\left(\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle },\\ A_{3} & \leq\left\Vert T\right\Vert _{\infty,2}\left(\sum_{l\in S_{C}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}K_{l}^{\oplus}\right\Vert _{\text{HS}}\right)\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \nonumber, \\ A_{4} & \leq\left\Vert T\right\Vert _{\infty,2}\left(\sum_{l\in S_{C}}\left\Vert K_{l}^{\oplus}\right\Vert _{\infty,2}\right)\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle ,\nonumber \end{align} $$

all of which are also accounted for by the statement.

7 Analysis of the one-body operators K, $A (t )$ and $B (t )$

In this section, we study the one-body operators on $\ell ^2(L_k)$ defined in Section 5, including $K_k$ introduced in (5.23) and $A_k,B_k$ defined in Proposition 5.7:

(7.1) $$ \begin{align} K_{k}&=-\frac{1}{2}\log\left(h_{k}^{-\frac{1}{2}}\big(h_{k}^{2}+2P_{h_{k}^{\frac{1}{2}}v_{k}}\big)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}\right), \nonumber\\ A_{k} (t ) & =\frac{1}{2}(e^{tK_{k}}(h_{k}+2P_{v_{k}})e^{tK_{k}}+e^{-tK_{k}}h_{k}e^{-tK_{k}})-h_{k},\nonumber\\ B_{k} (t ) & =\frac{1}{2}(e^{tK_{k}}(h_{k}+2P_{v_{k}})e^{tK_{k}}-e^{-tK_{k}}h_{k}e^{-tK_{k}}) , \end{align} $$

where

(7.2) $$ \begin{align} h_{k}e_{p} =\lambda_{k,p}e_{p},\quad\;\;\lambda_{k,p}=\frac{1}{2}\left(|p|^{2}-|p-k|^{2}\right), \quad P_{v_{k}} = |v_k \rangle \langle v_k|, \quad v_{k}=\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sum_{p\in L_{k}}e_{p}, \end{align} $$

and $(e_{p})_{p\in L_{k}}$ is the standard orthonormal basis of $\ell ^{2}(L_{k})$ . We will need precise estimates on these operators to control the quasi-bosonic Bogolubov transformation $e^{\mathcal {K}}$ diagonalizing the bosonizable terms. In particular, we will prove the following bounds.

Proposition 7.1 (Trace formulas).

For all $k\in \mathbb {Z}_{\ast }^{3}$ , it holds that $K_k\le 0$ and

$$ \begin{align*}\operatorname{\mathrm{tr}}\left(K_{k}\right) = -\frac{1}{4}\log\left(1+2\hat{V}_{k}\left(\frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p\in L_{k}}\lambda_{k,p}^{-1}\right) \right) \ge - C\hat{V}_{k}. \end{align*} $$

Moreover, with $E_k= e^{-K_k} h_k e^{-K_k}$ , we have

$$\begin{align*}\operatorname{\mathrm{tr}}\left(E_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|=\frac{1}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt, \end{align*}$$

with $F (x )=\log \left(1+x\right)-x$ , and

$$\begin{align*}\left|\operatorname{\mathrm{tr}}\left(E_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|\right|\leq Ck_{F}\hat{V}_{k}^{2}|k|,\quad k_{F}\rightarrow\infty. \end{align*}$$

Here, $C>0$ is a constant independent of k and $k_{F}$ .

Proposition 7.2 (Matrix element estimates).

For all $k\in \overline {B}\left(0,2k_{F}\right)\cap \mathbb {Z}_{\ast }^{3}$ , it holds that

$$ \begin{align*} \left\Vert K_{k}\right\Vert _{\infty,2} & \leq C\hat{V}_{k}\log\left(k_{F}\right)^{\frac{1}{3}}k_{F}^{-\frac{2}{3}}|k|^{1+\frac{5}{6}}, \end{align*} $$

and for all $t\in \left[0,1\right]$ , that

$$\begin{align*}\left\Vert A_{k} (t )\right\Vert _{\infty,2},\,\left\Vert B_{k} (t )\right\Vert _{\infty,2}\leq C\hat{V}_{k}|k|^{\frac{1}{2}}\left(1+\hat{V}_{k}\right). \end{align*}$$

Moreover, with $E_k= e^{-K_k} h_k e^{-K_k}$ , we have

$$ \begin{align*}\max_{p\in L_{k}}\left|\left\langle e_{p},\left(E_k -h_{k}\right)e_{p}\right\rangle \right| \leq Ck_{F}^{-1}\hat{V}_{k}\left(1+\hat{V}_{k}\right). \end{align*} $$

Here, $C>0$ is a constant independent of k and $k_{F}$ .

Proposition 7.3 (Kinetic estimates).

For all $k\in \overline {B}\left(0,2k_{F}\right)$ , it holds as $k_{F}\rightarrow \infty $ that

$$ \begin{align*} \left\Vert h_{k}^{-\frac{1}{2}}K_{k}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\hat{V}_{k}|k|^{3+\frac{2}{3}}\\ \left\Vert \left\{ K_{k},h_{k}\right\} h_{k}^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq Ck_{F}^{\frac{1}{2}}\hat{V}_{k}|k|^{\frac{1}{2}}\\ \left\Vert h_{k}^{-\frac{1}{2}}\left\{ K_{k},h_{k}\right\} h_{k}^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq C\hat{V}_{k}, \end{align*} $$

and for all $t\in \left[0,1\right]$ ,

$$\begin{align*}\max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}A_{k} (t )e_{p}\right\Vert ,\,\max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}B_{k} (t )e_{p}\right\Vert \leq Ck_{F}^{-\frac{1}{2}}\hat{V}_{k}\left(1+\hat{V}_{k}^{2}\right). \end{align*}$$

Here, $C>0$ is a constant independent of k and $k_{F}$ .

Notation. In order to simplify the notation, we will throughout this section let $h:V\rightarrow V$ denote any positive self-adjoint operator acting on an n-dimensional Hilbert space V, let $(x_{i})_{i=1}^{n}$ be an eigenbasis for h with eigenvalues $\left(\lambda _{i}\right)_{i=1}^{n}$ and let $v\in V$ be any vector satisfying $\left\langle x_{i},v\right\rangle \geq 0$ for all $1\leq i\leq n$ . We will establish general results for the operators (c.f. (7.1))

(7.3) $$ \begin{align} K&=-\frac{1}{2}\log\big(h^{-\frac{1}{2}}\big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)^{\frac{1}{2}}h^{-\frac{1}{2}}\big),\\ A\big(t\big) & =\frac{1}{2}\big(e^{tK}\big(h+2P_{v}\big)e^{tK}+e^{-tK}he^{-tK}\big)-h,\nonumber\\ B\big(t\big) & =\frac{1}{2}\big(e^{tK}\big(h+2P_{v}\big)e^{tK}-e^{-tK}he^{-tK}\big)\nonumber \end{align} $$

and then at the end insert the specific choice (7.2) to get explicit estimates.

We will prove the trace formulas first. Then we derive general estimates for the matrix elements of the operators $e^{-2K}$ and $e^{2K}$ in terms of a single, simpler operator T. This allows us to show that all matrix elements of K are non-negative, which in turn implies that all matrix elements of $e^{-tK}$ , $\sinh \left(-tK\right)$ and $\cosh \left(-tK\right)$ are convex with respect to t. With these estimates, we can then obtain the desired estimates of K, $A(t)$ and $B(t)$ .

7.1 Trace formulas

In this section, we prove Proposition 7.1. We will prove some general results using the notation in (7.3), and then we insert the special choice of $h_k$ , $v_k$ in (7.2) to conclude. Let us start with the following:

Proposition 7.4. The operator K in (7.3) satisfies $K\le 0$ and

$$\begin{align*}\operatorname{\mathrm{tr}}(K)=-\frac{1}{4}\log(1+2\langle v,h^{-1}v \rangle ). \end{align*}$$

Proof. Since $h^{2}+2P_{h^{\frac {1}{2}}v} \ge h^2>0$ and $A\mapsto A^{\frac {1}{2}}$ is operator monotone, we find that

(7.4) $$ \begin{align} h^{-\frac{1}{2}}\big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)^{\frac{1}{2}}h^{-\frac{1}{2}}\geq h^{-\frac{1}{2}}(h^{2})^{\frac{1}{2}}h^{-\frac{1}{2}}=1. \end{align} $$

Hence, K is well defined and $K\le 0$ . By the identity $\text {tr}\left(\log (A)\right)=\log \left(\det (A)\right)$ and multiplicativity of the determinant, we find

(7.5) $$ \begin{align} \text{tr}\big(K\big) & =-\frac{1}{2}\log\big(\det\big(h^{-\frac{1}{2}}\big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)^{\frac{1}{2}}h^{-\frac{1}{2}}\big)\big)\nonumber\\ &=-\frac{1}{4}\log\big(\det\big(h\big)^{-1}\det\big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)\det\big(h\big)^{-1}\big)\nonumber \\ & =-\frac{1}{4}\log\big(\det\big(h^{-1}\big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)h^{-1}\big)\big)=-\frac{1}{4}\log\big(\det\big(1+2P_{h^{-\frac{1}{2}}v}\big)\big), \end{align} $$

and by Sylvester’s determinant theorem [40], $\det \left(1+\alpha P_{x}\right)=1+\alpha \left\Vert x\right\Vert ^{2}$ for any $\alpha \in \mathbb {C}$ ; hence,

(7.6) $$ \begin{align} \text{tr}(K)=-\frac{1}{4}\log\big(1+2\Vert h^{-\frac{1}{2}}v\Vert ^{2}\big)=-\frac{1}{4}\log\big(1+2\langle v,h^{-1}v\rangle \big). \end{align} $$

Another exact trace formula which we will need is the following integral representation of the square root of a rank one perturbation, first presented in [5].

Proposition 7.5. Let $\left(H,\left\langle \cdot ,\cdot \right\rangle \right)$ be a Hilbert space and let $A:H\rightarrow H$ be a positive self-adjoint operator. Then for any $x\in H$ and $g\in \mathbb {R}$ such that $A+gP_{x}>0$ , it holds that

$$\begin{align*}\left(A+gP_{x}\right)^{\frac{1}{2}}=A^{\frac{1}{2}}+\frac{2g}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+g\left\langle x,\left(A+t^{2}\right)^{-1}x\right\rangle }P_{\left(A+t^{2}\right)^{-1}x}\,dt \end{align*}$$

and

$$\begin{align*}\operatorname{\mathrm{tr}}\left(\left(A+gP_{x}\right)^{\frac{1}{2}}\right)=\operatorname{\mathrm{tr}}\left(A^{\frac{1}{2}}\right)+\frac{1}{\pi}\int_{0}^{\infty}\log\left(1+g\left\langle x,\left(A+t^{2}\right)^{-1}x\right\rangle \right)dt. \end{align*}$$

Note that Proposition 7.5 follows from the Sherman–Morrison formula [39]

(7.7) $$ \begin{align} \left(A+gP_{x,y}\right)^{-1}=A^{-1}-\frac{g}{1+g\left\langle x,A^{-1}y\right\rangle }P_{\left(A^{\ast}\right)^{-1}x,A^{-1}y}, \end{align} $$

with $P_{x,y}= |y\rangle \langle x| = \left\langle x,\cdot \right\rangle y$ , and the functional calculus

(7.8) $$ \begin{align} \sqrt{A}=\frac{2}{\pi}\int_{0}^{\infty}\frac{A}{A+t^{2}}\,dt=\frac{2}{\pi}\int_{0}^{\infty}\left(1-\frac{t^{2}}{A+t^{2}}\right)dt \end{align} $$

for every self-adjoint non-negative operator A. Using this, we conclude the following:

Proposition 7.6. The trace of $E-h$ where $E=e^{-K}he^{-K}$ is given by

$$\begin{align*}\operatorname{\mathrm{tr}}\left(E-h\right)=\frac{1}{\pi}\int_{0}^{\infty}\log\left(1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle \right)dt. \end{align*}$$

Proof. By cyclicity of the trace and the definition of K,

(7.9) $$ \begin{align} \text{tr}\left(e^{-K}he^{-K}\right)=\text{tr}\left(he^{-2K}\right)=\text{tr}\left(h\left(h^{-\frac{1}{2}}\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{2}}h^{-\frac{1}{2}}\right)\right)=\text{tr}\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{2}}, \end{align} $$

so applying Proposition 7.5 with $A=h^{2}$ , $x=h^{\frac {1}{2}}v$ and $g=2$ , we get the claim.

Proof of Proposition 7.1.

By inserting $h_{k}$ and $v_{k}$ in Proposition 7.4, we get $K_k\le 0$ and

(7.10) $$ \begin{align} \operatorname{\mathrm{tr}}\left(K_k\right)=-\frac{1}{4}\log\left(1+2\left\langle v_k,h_k^{-1}v_k \right\rangle \right). \end{align} $$

With the choice of $h_{k}$ and $v_{k}$ in (7.2), we have

(7.11) $$ \begin{align} 0\le \left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle =\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p,q\in L_{k}}\left\langle e_{p},h_{k}^{-1}e_{q}\right\rangle =\frac{\hat{V}_{k} k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p\in L_{k}}\lambda_{k,p}^{-1} \le C \hat V_k, \end{align} $$

where the last inequality is taken from Proposition A.2 in the Appendix. Combining with the bound $ \log (1+x) \le x $ with $x>0$ , we find that

(7.12) $$ \begin{align} \operatorname{\mathrm{tr}}\left(K_k\right)=-\frac{1}{4}\log\left(1+2 \hat{V}_{k}\left(\frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p\in L_{k}}\lambda_{k,p}^{-1}\right) \right) \ge - C \hat V_k. \end{align} $$

Next, using Proposition 7.6 and the identity (c.f. (7.8))

(7.13) $$ \begin{align} \left|L_{k}\right|=\sum_{p\in L_{k}}1=\frac{2}{\pi}\int_{0}^{\infty}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\,dt, \end{align} $$

we conclude that

(7.14) $$ \begin{align} \operatorname{\mathrm{tr}}\left(E_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|=\frac{1}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt \end{align} $$

with $F (x )=\log \left(1+x\right)-x$ . Since $\left|F (x )\right|\leq \frac {1}{2}x^{2}$ , we have

(7.15) $$ \begin{align} \left|\text{tr}\left(E_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|\right| & \leq\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{2}\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)^{2}dt\\ & =\frac{\hat{V}_{k}^{2}k_{F}^{-2}}{(2\pi)^{7}}\sum_{p,q\in L_{k}}\int_{0}^{\infty}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\frac{\lambda_{k,q}}{\lambda_{k,q}^{2}+t^{2}}\,dt\nonumber, \end{align} $$

and by the integral identity

(7.16) $$ \begin{align} \int_{0}^{\infty}\frac{a}{a^{2}+t^{2}}\frac{b}{b^{2}+t^{2}}\,dt=\frac{\pi}{2}\frac{1}{a+b},\quad a,b>0, \end{align} $$

it holds that

(7.17) $$ \begin{align} \sum_{p,q\in L_{k}}\int_{0}^{\infty}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\frac{\lambda_{k,q}}{\lambda_{k,q}^{2}+t^{2}}\,dt &=\frac{\pi}{2}\sum_{p,q\in L_{k}}\frac{1}{\lambda_{k,p}+\lambda_{k,q}}\leq\frac{\pi}{2}\sum_{p,q\in L_{k}}\frac{1}{\sqrt{\lambda_{k,p}}\sqrt{\lambda_{k,q}}} \nonumber \\ &=\frac{\pi}{2}\left(\sum_{p\in L_{k}}\lambda_{k,p}^{-\frac{1}{2}}\right)^{2}. \end{align} $$

By Proposition A.1, we have for any $k\in \mathbb {Z}_{\ast }^{3}$ that

(7.18) $$ \begin{align} \sum_{p\in L_{k}}\lambda_{k,p}^{-\frac{1}{2}}\leq C\begin{cases} k_{F}^{\frac{3}{2}}\sqrt{|k|} & |k|<2k_{F}\\ k_{F}^{3}|k|^{-1} & |k|\geq2k_{F} \end{cases}\leq Ck_{F}^{\frac{3}{2}}\sqrt{|k|},\quad k_{F}\rightarrow\infty \end{align} $$

for a constant $C>0$ independent of k and $k_{F}$ , so we get the desired bound

(7.19) $$ \begin{align} \left|\text{tr}\left(E_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|\right|\leq C\hat{V}_{k}^{2}k_{F}^{-2}\left(k_{F}^{\frac{3}{2}}\sqrt{|k|}\right)^{2}=Ck_{F}\hat{V}_{k}^{2}|k|. \end{align} $$

7.2 Preliminary estimates for $e^{-2K}$ and $e^{2K}$

The square root formula also yields the following exact representations of $e^{-2K}$ and $e^{2K}$ :

Proposition 7.7. The operator K in (7.3) satisfies

$$ \begin{align*} e^{-2K} & =1+\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h \left(h^{2}+t^{2}\right)^{-1}v\right\rangle }P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt\\ e^{2K} & =1-\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v\right\rangle t^{2}}P_{h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v}\,dt. \end{align*} $$

Proof. Let us consider

(7.20) $$ \begin{align} e^{-2K}=h^{-\frac{1}{2}}\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{2}}h^{-\frac{1}{2}}. \end{align} $$

first. Applying Proposition 7.5 with $A=h^{2}$ , $x=h^{\frac {1}{2}}v$ and $g=2$ , again we find

(7.21) $$ \begin{align} \big(h^{2}+2P_{h^{\frac{1}{2}}v}\big)^{\frac{1}{2}} & =(h^{2})^{\frac{1}{2}}+\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\langle h^{\frac{1}{2}}v,(h^{2}+t^{2})^{-1}h^{\frac{1}{2}}v\rangle }P_{\left(h^{2}+t^{2}\right)^{-1}h^{\frac{1}{2}}v}\,dt \nonumber\\ & =h+\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\langle v,h(h^{2}+t^{2})^{-1}v\rangle }P_{h^{\frac{1}{2}}\left(h^{2}+t^{2}\right)^{-1}v}\,dt \end{align} $$

whence

(7.22) $$ \begin{align} e^{-2K} & =h^{-\frac{1}{2}}\left(h+\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }P_{h^{\frac{1}{2}}\left(h^{2}+t^{2}\right)^{-1}v}\,dt\right)h^{-\frac{1}{2}}\nonumber\\ & =1+\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt. \end{align} $$

For $e^{2K}=h^{\frac {1}{2}}\big(h^{2}+2P_{h^{\frac {1}{2}}v}\big)^{-\frac {1}{2}}h^{\frac {1}{2}}$ , we first use (7.7) to write

(7.23) $$ \begin{align} (h^{2}+2P_{h^{\frac{1}{2}}v})^{-1} &=(h^{2})^{-1}-\frac{2}{1+2\left\langle h^{\frac{1}{2}}v,\left(h^{2}\right)^{-1}h^{\frac{1}{2}}v\right\rangle }P_{\left(h^{2}\right)^{-1}h^{\frac{1}{2}}v} \nonumber \\ &=h^{-2}-\frac{2}{1+2\left\langle v,h^{-1}v\right\rangle }P_{h^{-\frac{3}{2}}v}. \end{align} $$

As this is an equality, the right-hand side is, in fact, positive (as the left-hand side is), so we may apply Proposition 7.5 with $A=h^{-2}$ , $x=h^{-\frac {3}{2}}v$ and $g=-2\left(1+2\left\langle v,h^{-1}v\right\rangle \right)^{-1}$ for

(7.24) $$ \begin{align} & \quad\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{-\frac{1}{2}}=\left(h^{-2}-\frac{2}{1+2\left\langle v,h^{-1}v\right\rangle }P_{h^{-\frac{3}{2}}v}\right)^{\frac{1}{2}}\nonumber \\ & =\left(h^{-2}\right)^{\frac{1}{2}}-\frac{2}{1+2\left\langle v,h^{-1}v\right\rangle }\frac{2}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1-\frac{2}{1+2\left\langle v,h^{-1}v\right\rangle }\left\langle h^{-\frac{3}{2}}v,\left(h^{-2}+t^{2}\right)^{-1}h^{-\frac{3}{2}}v\right\rangle }P_{\left(h^{-2}+t^{2}\right)^{-1}h^{-\frac{3}{2}}v}\,dt\nonumber \\ & =h^{-1}-\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h^{-1}v\right\rangle -2\left\langle v,h^{-3}\left(h^{-2}+t^{2}\right)^{-1}v\right\rangle }P_{h^{-\frac{3}{2}}\left(h^{-2}+t^{2}\right)^{-1}v}\,dt\nonumber\\ & =h^{-1}-\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v\right\rangle t^{2}}P_{h^{-\frac{3}{2}}\left(h^{-2}+t^{2}\right)^{-1}v}\,dt. \end{align} $$

Hence,

(7.25) $$ \begin{align} e^{2K} & =h^{\frac{1}{2}}\left(h^{-1}-\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v\right\rangle t^{2}}P_{h^{-\frac{3}{2}}\left(h^{-2}+t^{2}\right)^{-1}v}\,dt\right)h^{\frac{1}{2}}\\ & =1-\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v\right\rangle t^{2}}P_{h^{-1}\left(h^{-2}+t^{2}\right)^{-1}v}\,dt.\nonumber \end{align} $$

These exact formulas now allow us to derive some simple estimates for $e^{-2K}-1$ and $1-e^{2K}$ . To state these estimates, we first define a new operator T on $\ell ^2 (L_k)$ with matrix elements

(7.26) $$ \begin{align} \left\langle x_{i},T x_{j}\right\rangle =2\frac{\left\langle x_{i},v \right\rangle \left\langle v, x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}},\quad \forall 1\le i,j \le n. \end{align} $$

Recall that $(x_i)_{i=1}^n$ are an eigenbasis of h with eigenvalues $\lambda _{i}$ ’s and $\langle x_i,v\rangle \ge 0$ for all $1\le i\le n$ .

Proposition 7.8. For K in (7.3) and T in (7.26), we have both the operator estimates

$$ \begin{align*} 0&\leq e^{-2K} -1 \le T \leq ( 1+2\langle v ,h^{-1}v\rangle ) (e^{-2K}-1),\\ 0&\leq 1- e^{2K} \le T \leq ( 1+2\langle v ,h^{-1}v\rangle ) (1-e^{2K}), \end{align*} $$

and for all $1\le i,j \le n$ , the elementwise estimates

$$ \begin{align*} 0&\leq \langle x_{i}, (e^{-2K} -1) x_j \rangle \le \langle x_{i}, T x_j \rangle \le ( 1+2\langle v ,h^{-1}v\rangle ) \langle x_{i},(e^{-2K}-1)x_{j}\rangle,\\ 0&\leq \langle x_{i}, (1-e^{2K}) x_j \rangle \le \langle x_{i}, T x_j \rangle \le ( 1+2\langle v ,h^{-1}v\rangle ) \langle x_{i},( 1-e^{2K})x_{j}\rangle. \end{align*} $$

Proof. We first prove the bound $0\le e^{-2K}-1\leq T$ . Obviously, $0\le e^{-2K}-1$ since $K\le 0$ . Noting that $\langle v,h(h^{2}+t^{2})^{-1}v\rangle \geq 0$ and $P_{\left(h^{2}+t^{2}\right)^{-1}v}\geq 0$ for all $t\in \left[0,\infty \right)$ , we have by the first identity of Proposition 7.7 that

(7.27) $$ \begin{align} e^{-2K}-1=\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt\leq\frac{4}{\pi}\int_{0}^{\infty}t^{2}P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt. \end{align} $$

We claim that the right-hand side is precisely T. To see this, we compute the matrix elements with respect to $(x_{i})_{i=1}^{n}$ : For any $1\leq i,j\leq n$ , we have

(7.28) $$ \begin{align} & \left\langle x_{i},\left(\frac{4}{\pi}\int_{0}^{\infty}t^{2}P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt\right)x_{j}\right\rangle =\frac{4}{\pi}\int_{0}^{\infty}t^{2}\left\langle x_{i},\left(h^{2}+t^{2}\right)^{-1}v\right\rangle \left\langle \left(h^{2}+t^{2}\right)^{-1}v,x_{j}\right\rangle dt\nonumber \\ & =\frac{4}{\pi}\int_{0}^{\infty}t^{2}\frac{\left\langle x_{i},v\right\rangle }{\lambda_{i}^{2}+t^{2}}\frac{\left\langle v,x_{j}\right\rangle }{\lambda_{j}^{2}+t^{2}}\,dt=\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left(\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{\left(\lambda_{i}^{2}+t^{2}\right)\left(\lambda_{j}^{2}+t^{2}\right)}\,dt\right)\nonumber\\ & =\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left(\frac{4}{\pi}\frac{\pi}{2}\frac{1}{\lambda_{i}+\lambda_{j}}\right)=2\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}=\left\langle x_{i},Tx_{j}\right\rangle , \end{align} $$

where we used that $(x_{i})_{i=1}^{n}$ is an eigenbasis for h as well as the integral identity (7.16).

The lower bound $T\leq \left(1+2\left\langle v,h^{-1}v\right\rangle \right) (e^{-2K}-1)$ follows by the same argument as

(7.29) $$ \begin{align} \big\langle v,h\big(h^{2}+t^{2}\big)^{-1}v\big\rangle \leq\big\langle v,h\big(h^{2}\big)^{-1}v\big\rangle =\big\langle v,h^{-1}v\big\rangle \end{align} $$

for all $t\in \left[0,\infty \right)$ , so

(7.30) $$ \begin{align} e^{-2K}-1\geq\frac{1}{1+2\left\langle v,h^{-1}v\right\rangle }\left(\frac{4}{\pi}\int_{0}^{\infty}t^{2}P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt\right)=\frac{1}{1+2\left\langle v,h^{-1}v\right\rangle }T. \end{align} $$

The bounds

(7.31) $$ \begin{align} 0\le 1-e^{2K}\leq T \le \big(1+2\big\langle v,h^{-1}v\big\rangle \big) (1-e^{2K}) \end{align} $$

follow by exactly the same argument, starting from the second identity of Proposition 7.7, using that

(7.32) $$ \begin{align} 0\leq\big\langle v,h^{-1}\big(h^{-2}+t^{2}\big)^{-1}v\big\rangle t^{2}\leq\big\langle v,h^{-1}v\big\rangle \end{align} $$

for all $t\in \left[0,\infty \right)$ as well as the integral identity (7.16).

The matrix element estimates likewise follow by the same argument as, for example,

(7.33) $$ \begin{align} 0&\le \;\left\langle x_{i},\left(e^{-2K}-1\right)x_{j}\right\rangle =\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }\frac{\left\langle x_{i},v\right\rangle }{\lambda_{i}^{2}+t^{2}}\frac{\left\langle v,x_{j}\right\rangle }{\lambda_{j}^{2}+t^{2}}\,dt\nonumber \\ & =\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left(\frac{4}{\pi}\int_{0}^{\infty}\frac{1}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }\frac{t^{2}}{\left(\lambda_{i}^{2}+t^{2}\right)\left(\lambda_{j}^{2}+t^{2}\right)}\,dt\right)\\ & \leq\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left(\frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{\left(\lambda_{i}^{2}+t^{2}\right)\left(\lambda_{j}^{2}+t^{2}\right)}\,dt\right)=2\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}=\left\langle x_{i},Tx_{j}\right\rangle \nonumber \end{align} $$

by the assumption that the inner products $\left\langle x_{i},v\right\rangle $ and $\left\langle v,x_{j}\right\rangle $ are non-negative.

Remark 7.1 (Optimality of the estimates).

We may observe that the estimates for $e^{-2K}$ , $e^{2K}$ are, in general, optimal. To see this, let us add a small parameter $g\geq 0$ to the problem by substituting $\sqrt {g}v$ for v in equation (7.3) – that is, defining

(7.34) $$ \begin{align} K_{g}=-\frac{1}{2}\log\big(h^{-\frac{1}{2}}\big(h^{2}+2gP_{h^{\frac{1}{2}}v}\big)^{\frac{1}{2}}h^{-\frac{1}{2}}\big),\quad T_{g}=gT. \end{align} $$

Then the general bounds of the corollary read for $K_{g}$ that

(7.35) $$ \begin{align} \frac{1}{1+2g\left\langle v,h^{-1}v\right\rangle }T_{g}\leq-2K_{g}\leq T_{g}. \end{align} $$

Hence,

(7.36) $$ \begin{align} 0\geq-2K_{g}-T_{g}\geq-\left(1-\frac{1}{1+2g\left\langle v,h^{-1}v\right\rangle }\right)T_{g}=-\left(\frac{2\left\langle v,h^{-1}v\right\rangle }{1+2g\left\langle v,h^{-1}v\right\rangle }T\right)g^{2}\geq-Cg^{2}, \end{align} $$

which by self-adjointness of the operators involved implies that

(7.37) $$ \begin{align} -2K_{g}=T_{g}+O\big(g^{2}\big)=gT+O\big(g^{2}\big) \end{align} $$

with respect to, say, operator norm. This shows that the operator $T_{g}=gT$ is, in fact, the first-order expansion of $K_{g}$ with respect to the parameter g, which is then also the case for $e^{-2K_{g}}-1$ , $1-e^{2K_{g}}$ as, for example, $e^{-2K_{g}}-1=-2K_{g}+O\left(g^{2}\right)=T_{g}+O\left(g^{2}\right)$ . The estimate

(7.38) $$ \begin{align} \big\langle x_{i},\big(e^{-2K_{g}}-1\big)x_{j}\big\rangle \leq\big\langle x_{i},T_{g}x_{j}\big\rangle \end{align} $$

is therefore (asymptotically) optimal since $T_{g}$ is precisely the small g limit of $e^{-2K_{g}}-1$ .

This is relevant for our application, for although we do not have an explicit parameter g to consider, we do have $\hat V_k$ as an effective one. More precisely, the summability condition of $\hat V_k$ ensures that essentially all but finitely many coefficients $\hat {V}_{k}$ are small, even when the coefficients $(\hat {V}_{k})_{k\in \mathbb {Z}^{3}}$ are not finitely supported.

7.3 Matrix element estimates for K, $A(t)$ , $B(t)$

In this section, we prove Proposition 7.1. As before, we will prove some general results using the notation from (7.3), and then we insert $h_k$ , $v_k$ from (7.2) at the end. Recall that $(x_i)_{i}$ is an eigenbasis of h. We start with the following:

Proposition 7.9. For all $1\leq i,j\leq n$ , we have $\langle x_i, -K x_j \rangle \ge 0$ and the functions

$$\begin{align*}t\mapsto\big\langle x_{i},\big(e^{-tK}-1\big)x_{j}\big\rangle ,\,\left\langle x_{i},\sinh\left(-tK\right)x_{j}\right\rangle ,\,\left\langle x_{i},\left(\cosh\left(-tK\right)-1\right)x_{j}\right\rangle \end{align*}$$

are non-negative and convex for $t\in \left[0,\infty \right)$ .

Proof. By Proposition 7.8, the operator $S=1-e^{2K}$ satisfies that $0\le S<1$ and $\langle x_i, S x_j\rangle \ge 0$ for all $1\le i,j\le n$ . By writing

(7.39) $$ \begin{align} -2K = - \log(1-S)= S+ \frac{S^2}{2}+ \frac{S^3}{3}+ \frac{S^4}{4} + ..., \end{align} $$

we find that $-2K$ also has non-negative matrix elements. By using the series expansion again, we see that for any $1\leq i,j\leq n$ and $t\in \left[0,\infty \right)$ ,

(7.40) $$ \begin{align} \big\langle x_{i},\big(e^{-tK}-1\big)x_{j}\big\rangle & =\sum_{m=1}^{\infty}\frac{\big\langle x_{i},(-K)^{m}x_{j}\big\rangle }{m!}t^{m}\geq0,\nonumber\\ \frac{d^{2}}{dt^{2}}\big\langle x_{i},\big(e^{-tK}-1\big)x_{j}\big\rangle & =\sum_{m=3}^{\infty}\frac{\big\langle x_{i},(-K)^{m}x_{j}\big\rangle }{\left(m-2\right)!}t^{m-2}\geq0, \end{align} $$

yielding the claim for $t\mapsto \left\langle x_{i},\left(e^{-tK}-1\right)x_{j}\right\rangle $ . The functions $ t\mapsto \left\langle x_{i},\sinh \left(-tK\right)x_{j}\right\rangle $ and $t\mapsto \left\langle x_{i},\left(\cosh \left(-tK\right)-1\right)x_{j}\right\rangle $ can be treated similarly.

Next, we have the key matrix element bounds.

Proposition 7.10. For all $1\leq i,j\leq n$ and $t\in \left[0,1\right]$ , we have the elementwise estimates

$$ \begin{align*} & \left|\left\langle x_{i},Kx_{j}\right\rangle \right|,\,\left|\left\langle x_{i},\left(e^{-tK}-1\right)x_{j}\right\rangle \right|,\,\left|\left\langle x_{i},\left(1-e^{tK}\right)x_{j}\right\rangle \right|,\\ & \qquad\;\;\left|\left\langle x_{i},\sinh\left(-tK\right)x_{j}\right\rangle \right|,\,\left|\left\langle x_{i},\left(\cosh\left(-tK\right)-1\right)x_{j}\right\rangle \right|\leq \frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}. \end{align*} $$

Proof. The arguments for $e^{-tK}-1$ , $\sinh \left(-tK\right)$ and $\cosh \left(-tK\right)-1$ are again the same, so we focus on $e^{-tK}-1$ . By the convexity of Proposition 7.9 and the elementwise estimate of Proposition 7.8, we find for all $t\in \left[0,1\right]$ that

(7.41) $$ \begin{align} 0 & \leq\left\langle x_{i},\left(e^{-tK}-1\right)x_{j}\right\rangle \leq\left(1-\frac{t}{2}\right)\left\langle x_{i},\left(e^{-0\cdot K}-1\right)x_{j}\right\rangle +\frac{t}{2}\left\langle x_{i},\left(e^{-2K}-1\right)x_{j}\right\rangle \nonumber\\ & =\frac{t}{2}\left\langle x_{i},\left(e^{-2K}-1\right)x_{j}\right\rangle \leq\frac{1}{2}\left|\left\langle x_{i},\left(e^{-2K}-1\right)x_{j}\right\rangle \right|\leq\frac{1}{2}\left\langle x_{i},Tx_{j}\right\rangle =\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}. \end{align} $$

This also gives us the estimate for K as

(7.42) $$ \begin{align} 0\leq\left\langle x_{i},\left(-K\right)x_{j}\right\rangle \leq\sum_{m=1}^{\infty}\frac{1}{m!}\left\langle x_{i},\left(-K\right)^{m}x_{j}\right\rangle =\left\langle x_{i},\left(e^{-K}-1\right)x_{j}\right\rangle \leq\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}, \end{align} $$

where we used again the positivity of $\langle x_i, -K x_j\rangle $ from Proposition 7.9. Finally, the estimate for $1-e^{-tK}$ is deduced from that of $\sinh \left(-tK\right)$ and $\cosh \left(-tK\right)-1$ as

(7.43) $$ \begin{align} |\langle x_{i},(1-e^{tK})x_{j}\rangle | & =\left|\left\langle x_{i},\sinh\left(-tK\right)x_{j}\right\rangle -\left\langle x_{i},\left(\cosh\left(-tK\right)-1\right)x_{j}\right\rangle \right|\nonumber\\ & \leq\max\left\{ \left|\left\langle x_{i},\sinh\left(-tK\right)x_{j}\right\rangle \right|,\left|\left\langle x_{i},\left(\cosh\left(-tK\right)-1\right)x_{j}\right\rangle \right|\right\} \leq\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}} , \end{align} $$

where we also used the positivity of $\left\langle x_{i},\left(\cosh \left(-tK\right)-1\right)x_{j}\right\rangle $ and $\left\langle x_{i},\sinh \left(-tK\right)x_{j}\right\rangle $ from Proposition 7.9 to justify the first inequality.

As a simple application of these estimates, we can easily obtain the following:

Proposition 7.11. It holds that

$$\begin{align*}\left\Vert K\right\Vert _{\infty,2}\leq\alpha\sqrt{\left\langle v,h^{-2}v\right\rangle }, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ .

Proof. We estimate using Proposition 7.10 that

(7.44) $$ \begin{align} \left\Vert K\right\Vert _{\infty,2}^{2} & =\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},Kx_{j}\right\rangle \right|^{2}\leq\sum_{i=1}^{n}\max_{1\leq j\leq n}\left(\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right)^{2} \nonumber\\ & \leq\left(\max_{1\leq j\leq n}\left|\left\langle v,x_{j}\right\rangle \right|^{2}\right)\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}^{2}}=\alpha^{2}\left\langle v,h^{-2}v\right\rangle. \end{align} $$

Now we consider $A (t )$ and $B (t )$ , which can be written as

(7.45) $$ \begin{align} A (t )=A_{h} (t )+e^{tK}P_{v}e^{tK},\quad B (t )=B_{h} (t )+e^{tK}P_{v}e^{tK}, \end{align} $$

for

(7.46) $$ \begin{align} A_{h} (t ) & =\frac{1}{2}\left(e^{tK}he^{tK}+e^{-tK}he^{-tK}\right)-h\nonumber \\ & =\cosh\left(-tK\right)h\cosh\left(-tK\right)+\sinh\left(-tK\right)h\sinh\left(-tK\right)-h \nonumber\\ & =\sinh\left(-tK\right)h\sinh\left(-tK\right)+\left(\cosh\left(-tK\right)-1\right)h\left(\cosh\left(-tK\right)-1\right)+\left\{ h,\cosh\left(-tK\right)-1\right\} \end{align} $$

and

(7.47) $$ \begin{align} B_{h} (t ) & =\frac{1}{2}\left(e^{tK}he^{tK}-e^{-tK}he^{-tK}\right)\nonumber \\ & =-\left(\cosh\left(-tK\right)h\sinh\left(-tK\right)+\sinh\left(-tK\right)h\cosh\left(-tK\right)\right) \nonumber\\ & =-\left(\left(\cosh\left(-tK\right)-1\right)h\sinh\left(-tK\right)+\sinh\left(-tK\right)h\left(\cosh\left(-tK\right)-1\right)+\left\{ h,\sinh\left(-tK\right)\right\} \right). \end{align} $$

Specifically, we must estimate the $\left\Vert \cdot \right\Vert _{\infty ,2}$ norms of $A (t )$ and $B (t )$ with respect to $(x_{i})_{i=1}^{n}$ . We begin with the $e^{tK}P_{v}e^{tK}$ term:

Proposition 7.12. It holds for all $t\in \left[0,1\right]$ that

$$\begin{align*}\left\Vert e^{tK}P_{v}e^{tK}\right\Vert _{\infty,2}\leq\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\Vert v\right\Vert, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ .

Proof. We first observe that

(7.48) $$ \begin{align} \left\Vert e^{tK}P_{v}e^{tK}\right\Vert _{\infty,2}^{2} & =\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},e^{tK}P_{v}e^{tK}x_{j}\right\rangle \right|^{2}=\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},e^{tK}v\right\rangle \right|^{2}\left|\left\langle v,e^{tK}x_{j}\right\rangle \right|^{2} \nonumber\\ & =\left(\max_{1\leq j\leq n}\left|\left\langle v,e^{tK}x_{j}\right\rangle \right|^{2}\right)\left\Vert e^{tK}v\right\Vert ^{2}\leq\left(\max_{1\leq j\leq n}\left|\left\langle v,e^{tK}x_{j}\right\rangle \right|^{2}\right)\left\Vert v\right\Vert ^{2} , \end{align} $$

where we used that by monotonicity of $e^{x}$ and the fact that $K\leq 0$ , $\left\Vert e^{tK}v\right\Vert ^{2}=\left\langle v,e^{2tK}v\right\rangle \leq \left\Vert v\right\Vert ^{2}$ . For the remaining factor, we first write

(7.49) $$ \begin{align} \left\langle v,e^{tK}x_{j}\right\rangle =\left\langle v,x_{j}\right\rangle +\left\langle v,\left(e^{tK}-1\right)x_{j}\right\rangle =\left\langle v,x_{j}\right\rangle +\sum_{i=1}^{n}\left\langle v,x_{i}\right\rangle \left\langle x_{i},\left(e^{tK}-1\right)x_{j}\right\rangle ,\quad1\leq j\leq n \end{align} $$

and estimate using Proposition 7.10 that

(7.50) $$ \begin{align} \left|\sum_{i=1}^{n}\left\langle v,x_{i}\right\rangle \left\langle x_{i},\left(e^{tK}-1\right)x_{j}\right\rangle \right| & \leq\sum_{i=1}^{n}\left|\left\langle v,x_{i}\right\rangle \right|\left|\left\langle x_{i},\left(e^{tK}-1\right)x_{j}\right\rangle \right|\leq\sum_{i=1}^{n}\left|\left\langle v,x_{i}\right\rangle \right|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\nonumber \\ & \leq\left\langle v,x_{j}\right\rangle \sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}=\left\langle v,x_{j}\right\rangle \left\langle v,h^{-1}v\right\rangle ,\quad1\leq j\leq n. \end{align} $$

Hence,

(7.51) $$ \begin{align} \left|\left\langle v,e^{tK}x_{j}\right\rangle \right|\leq\left\langle v,x_{j}\right\rangle +\left|\sum_{i=1}^{n}\left\langle v,x_{i}\right\rangle \left\langle x_{i},\left(e^{tK}-1\right)x_{j}\right\rangle \right|\leq\left\langle v,x_{j}\right\rangle \left(1+\left\langle v,h^{-1}v\right\rangle \right),\quad1\leq j\leq n, \end{align} $$

so returning to equation (7.48), we conclude that

(7.52) $$ \begin{align} \left\Vert e^{tK}P_{v}e^{tK}\right\Vert _{\infty}^{2}\leq\left(\max_{1\leq j\leq n}\left|\left\langle v,x_{j}\right\rangle \left(1+\left\langle v,h^{-1}v\right\rangle \right)\right|^{2}\right)\left\Vert v\right\Vert ^{2}=\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\Vert v\right\Vert ^{2}, \end{align} $$

implying the claim.

For $A_{h} (t )$ and $B_{h} (t )$ , we estimate the matrix elements of the operators appearing in the equations (7.46) and (7.47):

Proposition 7.13. It holds for all $1\leq i,j\leq n$ and $t\in \left[0,1\right]$ that, for $C_{t}=\cosh \left(-tK\right)-1$ and $S_{t}=\sinh \left(-tK\right)$ ,

(7.53) $$ \begin{align} \left|\left\langle x_{i},C_{t}hC_{t}x_{j}\right\rangle \right|,\left|\left\langle x_{i},C_{t}hS_{t}x_{j}\right\rangle \right|,\left|\left\langle x_{i},S_{t}hS_{t}x_{j}\right\rangle \right|\leq\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left\langle v,h^{-1}v\right\rangle \end{align} $$

and

(7.54) $$ \begin{align} \left|\left\langle x_{i},\left\{ h,C_{t}\right\} x_{j}\right\rangle \right|,\left|\left\langle x_{i},\left\{ h,S_{t}\right\} x_{j}\right\rangle \right|\leq\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle. \end{align} $$

Proof. The arguments for the elements of the two groups are the same, so we focus on particular representatives. For the first, we have by the estimates of Proposition 7.10 that

(7.55) $$ \begin{align} \left|\left\langle x_{i},\sinh\left(-tK\right)h\sinh\left(-tK\right),x_{j}\right\rangle \right| & =\left|\sum_{k=1}^{n}\lambda_{k}\left\langle x_{i},\sinh\left(-tK\right),x_{k}\right\rangle \left\langle x_{k},\sinh\left(-tK\right),x_{j}\right\rangle \right|\nonumber \\ & \leq\sum_{k=1}^{n}\lambda_{k}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\nonumber \\ & \leq\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}} = \left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left\langle v,h^{-1}v\right\rangle \end{align} $$

and for the second, that

(7.56) $$ \begin{align} \left|\left\langle x_{i},\left\{ h,\sinh\left(-tK\right)\right\} ,x_{j}\right\rangle \right| & =\left(\lambda_{i}+\lambda_{j}\right)\left|\left\langle x_{i},\sinh\left(-tK\right),x_{j}\right\rangle \right| \nonumber\\ & \leq\left(\lambda_{i}+\lambda_{j}\right)\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}=\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle . \end{align} $$

We can now obtain the desired estimate:

Proposition 7.14. It holds for all $t\in \left[0,1\right]$ that

$$\begin{align*}\left\Vert A (t )\right\Vert _{\infty,2},\,\left\Vert B (t )\right\Vert _{\infty,2}\leq3\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\Vert v\right\Vert, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ .

Proof. Again, the arguments for $A (t )$ and $B (t )$ are the same, so we focus on $A (t )$ . Using that $\left\Vert \cdot \right\Vert _{\infty ,2}$ is indeed a norm, and hence obeys the triangle inequality, we have for any $t\in \left[0,1\right]$ that

(7.57) $$ \begin{align} \left\Vert A (t )\right\Vert _{\infty,2} & \leq\left\Vert e^{tK}P_{v}e^{tK}\right\Vert _{\infty,2}+\left\Vert A_{h} (t )\right\Vert _{\infty,2} \nonumber \\ &\leq\left\Vert e^{tK}P_{v}e^{tK}\right\Vert _{\infty,2}+\left\Vert \sinh\left(-tK\right)h\sinh\left(-tK\right)\right\Vert _{\infty,2}\nonumber \\ &\quad +\left\Vert \left(\cosh\left(-tK\right)-1\right)h\left(\cosh\left(-tK\right)-1\right)\right\Vert _{\infty,2}+\left\Vert \left\{ h,\cosh\left(-tK\right)-1\right\} \right\Vert _{\infty,2}. \end{align} $$

We estimate $\left\Vert \sinh \left(-tK\right)h\sinh \left(-tK\right)\right\Vert _{\infty ,2}$ using Proposition 7.13 as

(7.58) $$ \begin{align} \left\Vert \sinh\left(-tK\right)h\sinh\left(-tK\right)\right\Vert _{\infty,2}^{2} & =\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},\sinh\left(-tK\right)h\sinh\left(-tK\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \left\langle v,h^{-1}v\right\rangle \right|^{2}\\ & =\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{2}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}=\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{2}\left\Vert v\right\Vert ^{2},\nonumber \end{align} $$

the same bound holding also for $\left\Vert \left(\cosh \left(-tK\right)-1\right)h\left(\cosh \left(-tK\right)-1\right)\right\Vert _{\infty ,2}^{2}$ . We likewise find

(7.59) $$ \begin{align} \left\Vert \left\{ h,\cosh\left(-tK\right)-1\right\} \right\Vert _{\infty,2}^{2} & =\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},\left\{ h,\cosh\left(-tK\right)-1\right\} x_{j}\right\rangle \right|^{2}\\ & \leq\sum_{i=1}^{n}\max_{1\leq j\leq n}\left|\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \right|^{2}=\alpha^{2}\left\Vert v\right\Vert ^{2},\nonumber \end{align} $$

so recalling the estimate of Proposition 7.12, we conclude that

(7.60) $$ \begin{align} \left\Vert A (t )\right\Vert _{\infty,2}\leq\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\Vert v\right\Vert +2\alpha\left\langle v,h^{-1}v\right\rangle \left\Vert v\right\Vert +\alpha\left\Vert v\right\Vert \leq3\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\Vert v\right\Vert. \end{align} $$

Now we come to the last ingredient of Proposition 7.1.

Proposition 7.15. Let $E=e^{-K}he^{-K}$ . For all $1\leq i,j\leq n$ , it holds that

$$\begin{align*}\big|\big\langle x_{i},\big(E-h\big)x_{j}\big\rangle \big|\leq\big(1+\big\langle v,h^{-1}v\big\rangle \big)\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle. \end{align*}$$

Proof. Using the identity

(7.61) $$ \begin{align} e^{-K}he^{-K}-h=\big\{ h,e^{-K}-1\big\} +\big(e^{-K}-1\big)h\big(e^{-K}-1\big), \end{align} $$

we can write

(7.62) $$ \begin{align} \big\langle x_{i},\big(e^{-K}he^{-K}-h\big)x_{j}\big\rangle =\big(\lambda_{i}+\lambda_{j}\big)\big\langle x_{i},\big(e^{-K}-1\big)x_{j}\big\rangle +\big\langle x_{i},\big(e^{-K}-1\big)h\big(e^{-K}-1\big)x_{j}\big\rangle. \end{align} $$

We can apply Proposition 7.10 to estimate the first term of this equation as

(7.63) $$ \begin{align} \big|\big(\lambda_{i}+\lambda_{j}\big)\big\langle x_{i},\big(e^{-K}-1\big)x_{j}\big\rangle \big|\leq\big(\lambda_{i}+\lambda_{j}\big)\frac{\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle }{\lambda_{i}+\lambda_{j}}=\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle \end{align} $$

and the second term as

(7.64) $$ \begin{align} \big|\big\langle x_{i},\big(e^{-K}-1\big)h\big(e^{-K}-1\big)x_{j}\big\rangle \big| & =\left|\sum_{k=1}^{n}\lambda_{k}\big\langle x_{i},\big(e^{-K}-1\big)x_{k}\big\rangle \big\langle x_{k},\big(e^{-K}-1\big)x_{j}\big\rangle \right|\nonumber \\ & \leq\sum_{k=1}^{n}\lambda_{k}\frac{\big\langle x_{i},v\big\rangle \big\langle v,x_{k}\big\rangle }{\lambda_{i}+\lambda_{k}}\frac{\big\langle x_{k},v\big\rangle \big\langle v,x_{j}\big\rangle }{\lambda_{k}+\lambda_{j}}\\ & \leq\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle \sum_{k=1}^{n}\frac{\big|\big\langle x_{k},v\big\rangle \big|^{2}}{\lambda_{k}}=\big\langle v,h^{-1}v\big\rangle \big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle \nonumber, \end{align} $$

which implies the claim.

Proof of Proposition 7.2.

Now we insert $h_{k}$ and $v_{k}$ to conclude. Using Proposition 7.11, and noting that ‘ $\alpha $ ’ of our problem is simply the constant

(7.65) $$ \begin{align} \max_{p\in L_{k}}\left\langle v_{k},e_{p}\right\rangle =\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}, \end{align} $$

we find that

(7.66) $$ \begin{align} \left\Vert K_{k}\right\Vert _{\infty,2} &\leq\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sqrt{\left\langle v_{k},h_{k}^{-2}v_{k}\right\rangle }=\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}^{2}}} \nonumber\\ &\leq C\hat{V}_{k}k_{F}^{-1}\sqrt{\sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}^{2}}}. \end{align} $$

The desired upper bound

(7.67) $$ \begin{align} \left\Vert K_{k}\right\Vert _{\infty,2} & \leq C\hat{V}_{k}\log\left(k_{F}\right)^{\frac{1}{3}}k_{F}^{-\frac{2}{3}}|k|^{1+\frac{5}{6}} \end{align} $$

then follows from an estimate from Proposition A.3 in the Appendix:

$$\begin{align*}\sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}^{2}} \leq C|k|^{3+\frac{2}{3}}(\log k_{F})^{\frac{2}{3}}k_{F}^{\frac{2}{3}},\quad k_{F}\rightarrow\infty. \end{align*}$$

However, by Proposition 7.14 and (7.11), we conclude that

(7.68) $$ \begin{align} \left\Vert A_{k} (t )\right\Vert _{\infty,2},\,\left\Vert B_{k} (t )\right\Vert _{\infty,2} & \leq3\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}}\left(1+\left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle \right)\sqrt{\frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}}\left|L_{k}\right|} \nonumber\\ & \leq C\hat{V}_{k}k_{F}^{-1}\sqrt{\left|L_{k}\right|}\left(1+\hat{V}_{k}k_{F}^{-1}\sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}}\right) \leq C\hat{V}_{k}|k|^{\frac{1}{2}}\left(1+\hat{V}_{k}\right) , \end{align} $$

where we used

(7.69) $$ \begin{align} |L_k| \le C|k| k_F^2, \quad \sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}} \le C k_F \end{align} $$

from Proposition A.1 and Proposition A.2. Finally, from Proposition 7.15, we have

(7.70) $$ \begin{align} \max_{p\in L_{k}} \left|\left\langle e_p,\left(E_k-h_k\right)e_p\right\rangle \right| & \leq \left(1+\left\langle v_k,h_k^{-1}v_k\right\rangle \right) \sup_{p\in L_k} |\left\langle e_p,v\right\rangle|^2 \nonumber\\ &= \left(1+\hat{V}_{k}k_{F}^{-1}\sum_{p\in L_{k}}\frac{1}{\lambda_{k,p}}\right) \frac{\hat{V}_{k}k_{F}^{-1}}{2(2\pi)^{3}} \le Ck_{F}^{-1}\hat{V}_{k}\left(1+\hat{V}_{k}\right) , \end{align} $$

where we used (7.65) and (7.69) again in the last estimate.

7.4 Kinetic estimates

Now we prove Proposition 7.3. Again, let us start with the notation (7.3). We have the following:

Proposition 7.16. Under the notation (7.3), it holds that

$$ \begin{align*} \left\Vert h^{-\frac{1}{2}}K\right\Vert _{\operatorname{\mathrm{HS}}} & \leq\left\langle v,h^{-\frac{3}{2}}v\right\rangle ,\\ \left\Vert \left\{ K,h\right\} h^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq2\left\Vert v\right\Vert \sqrt{\left\langle v,h^{-1}v\right\rangle },\\ \left\Vert h^{-\frac{1}{2}}\left\{ K,h\right\} h^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq2\left\langle v,h^{-1}v\right\rangle. \end{align*} $$

Proof. Using Proposition 7.10, we estimate

(7.71) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}K\right\Vert _{\text{HS}}^{2}=\sum_{i,j=1}^{n}\frac{1}{\lambda_{i}}\left|\left\langle x_{i},Kx_{j}\right\rangle \right|^{2}\leq\sum_{i,j=1}^{n}\frac{1}{\lambda_{i}}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2}\leq\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{3}{2}}}\right)^{2}=\left\langle v,h^{-\frac{3}{2}}v\right\rangle ^{2}, \end{align} $$

and for $\left\Vert \left\{ K,h\right\} h^{-\frac {1}{2}}\right\Vert _{\text {HS}}$ use that $\left\Vert \left\{ K,h\right\} h^{-\frac {1}{2}}\right\Vert _{\text {HS}}\leq \left\Vert Kh^{\frac {1}{2}}\right\Vert _{\text {HS}}+\left\Vert hKh^{-\frac {1}{2}}\right\Vert _{\text {HS}}$ to estimate

(7.72) $$ \begin{align} \left\Vert Kh^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2} & =\sum_{i,j=1}^{n}\lambda_{j}\left|\left\langle x_{i},Kx_{j}\right\rangle \right|^{2}\leq\sum_{i,j=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\frac{\left|\left\langle x_{j},v\right\rangle \right|^{2}}{\lambda_{j}}=\left\Vert v\right\Vert ^{2}\left\langle v,h^{-1}v\right\rangle \nonumber\\ \left\Vert hKh^{-\frac{1}{2}}\right\Vert _{\text{HS}}^{2} & =\sum_{i,j=1}^{n}\frac{\lambda_{i}^{2}}{\lambda_{j}}\left|\left\langle x_{i},Kx_{j}\right\rangle \right|^{2}\leq\sum_{i,j=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\frac{\left|\left\langle x_{j},v\right\rangle \right|^{2}}{\lambda_{j}}=\left\Vert v\right\Vert ^{2}\left\langle v,h^{-1}v\right\rangle \end{align} $$

for the claimed $\left\Vert \left\{ K,h\right\} h^{-\frac {1}{2}}\right\Vert _{\text {HS}}\leq 2\left\Vert v\right\Vert \sqrt {\left\langle v,h^{-1}v\right\rangle }$ . We likewise have that

(7.73) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}\left\{ K,h\right\} h^{-\frac{1}{2}}\right\Vert _{\text{HS}}\leq\left\Vert h^{-\frac{1}{2}}Kh^{\frac{1}{2}}\right\Vert _{\text{HS}}+\left\Vert h^{\frac{1}{2}}Kh^{-\frac{1}{2}}\right\Vert _{\text{HS}}=2\left\Vert h^{-\frac{1}{2}}Kh^{\frac{1}{2}}\right\Vert _{\text{HS}}, \end{align} $$

so the bound

(7.74) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}Kh^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2}=\sum_{i,j=1}^{n}\frac{\lambda_{i}}{\lambda_{j}}\left|\left\langle x_{i},Kx_{j}\right\rangle \right|^{2}\leq\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\right)^{2}=\left\langle v,h^{-1}v\right\rangle ^{2} \end{align} $$

implies the final claim.

For $A (t )$ and $B (t )$ , we recall the decompositions (7.45)-(7.47). Recall also that $(x_i)_i$ is an eigenbasis of h and $\langle x_i,v\rangle \ge 0$ for all $1\le i\le n$ . We first estimate the $e^{tK}P_{v}e^{tK}$ term:

Proposition 7.17. For all $t\in \left[0,1\right]$ , it holds that

$$\begin{align*}\max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}e^{tK}P_{v}e^{tK}x_{j}\right\Vert \leq\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sqrt{\left\langle v,h^{-1}v\right\rangle }, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ .

Proof. We write $e^{tK}P_{v}e^{tK}$ as

(7.75) $$ \begin{align} e^{tK}P_{v}e^{tK}=P_{v}+\big(e^{tK}-1\big)P_{v}+P_{v}\big(e^{tK}-1\big)+\big(e^{tK}-1\big)P_{v}\big(e^{tK}-1\big) \end{align} $$

and estimate each term separately. By the definition of $P_{v}$ , the first term is simply

(7.76) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}P_{v}x_{j}\right\Vert =\left|\left\langle v,x_{j}\right\rangle \right|\left\Vert h^{-\frac{1}{2}}v\right\Vert \leq\alpha\sqrt{\left\langle v,h^{-1}v\right\rangle }. \end{align} $$

For the remaining terms, we use Proposition 7.10 to estimate that

(7.77) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}\left(e^{tK}-1\right)P_{v}x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\left\langle x_{i},\left(e^{tK}-1\right)x_{k}\right\rangle \left\langle x_{k},P_{v}x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle \right|^{2}\qquad\qquad\qquad\qquad \end{align} $$

(7.78) $$ \begin{align} \leq\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\right)^{3}\leq\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{3}, \end{align} $$

and

(7.79) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}P_{v}\left(e^{tK}-1\right)x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\left\langle x_{i},P_{v}x_{k}\right\rangle \left\langle x_{k},\left(e^{tK}-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle \frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2}\\ & \leq\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\right)^{3}\leq\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{3}\nonumber \end{align} $$

and

(7.80) $$ \begin{align} & \,\,\left\Vert h^{-\frac{1}{2}}\left(e^{tK}-1\right)P_{v}\left(e^{tK}-1\right)x_{j}\right\Vert ^{2}\nonumber\\ &=\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k,l=1}^{n}\left\langle x_{i},\left(e^{tK}-1\right)x_{k}\right\rangle \left\langle x_{k},P_{v}x_{l}\right\rangle \left\langle x_{l},\left(e^{tK}-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k,l=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{l}\right\rangle \frac{\left\langle x_{l},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{l}+\lambda_{j}}\right|^{2}\\ & \leq\left|\left\langle v,x_{j}\right\rangle \right|^{2}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left(\sum_{k,l=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\frac{\left|\left\langle x_{l},v\right\rangle \right|^{2}}{\lambda_{l}}\right)^{2}\leq\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{5}\nonumber, \end{align} $$

which imply the claim.

Finally, the full estimates on $A (t )$ and $B (t )$ are now easily obtained:

Proposition 7.18. It holds for all $t\in \left[0,1\right]$ that

$$\begin{align*}\max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}A (t )x_{j}\right\Vert ,\,\max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}B (t )x_{j}\right\Vert \leq2\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sqrt{\left\langle v,h^{-1}v\right\rangle }, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ .

Proof. The estimates for $A (t )$ and $B (t )$ are similar, so we focus on $A (t )$ . We have

(7.81) $$ \begin{align} &\left\Vert h^{-\frac{1}{2}}A (t )x_{j}\right\Vert \nonumber\\ &\quad \leq\left\Vert h^{-\frac{1}{2}}\sinh\left(-tK\right)h\sinh\left(-tK\right)x_{j}\right\Vert +\left\Vert h^{-\frac{1}{2}}\left(\cosh\left(-tK\right)-1\right)h\left(\cosh\left(-tK\right)-1\right)x_{j}\right\Vert \nonumber\\ &\qquad +\left\Vert h^{-\frac{1}{2}}\left\{ h,\cosh\left(-tK\right)-1\right\} x_{j}\right\Vert +\left\Vert h^{-\frac{1}{2}}e^{tK}P_{v}e^{tK}x_{j}\right\Vert , \end{align} $$

and by Proposition 7.13, we can estimate that

(7.82) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}\sinh\left(-tK\right)h\sinh\left(-tK\right)x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\left\langle x_{i},\sinh\left(-tK\right)h\sinh\left(-tK\right)x_{j}\right\rangle \right|^{2}\nonumber\\ & \leq\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left\langle v,h^{-1}v\right\rangle ^{2}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\leq\alpha^{2}\left\langle v,h^{-1}v\right\rangle ^{3}, \end{align} $$

the same estimate holding also for $\left\Vert h^{-\frac {1}{2}}\left(\cosh \left(-tK\right)-1\right)h\left(\cosh \left(-tK\right)-1\right)x_{j}\right\Vert $ , and

(7.83) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}\left\{ h,\cosh\left(-tK\right)-1\right\} x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\left\langle x_{i},\left\{ h,\cosh\left(-tK\right)-1\right\} x_{j}\right\rangle \right|^{2}\nonumber\\ & \leq\left|\left\langle v,x_{j}\right\rangle \right|^{2}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\leq\alpha^{2}\left\langle v,h^{-1}v\right\rangle . \end{align} $$

Inserting also the estimate of Proposition 7.17, we thus obtain

(7.84) $$ \begin{align} \max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}A (t )x_{j}\right\Vert & \leq2\alpha\left\langle v,h^{-1}v\right\rangle ^{\frac{3}{2}}+\alpha\sqrt{\left\langle v,h^{-1}v\right\rangle }+\alpha(1+\left\langle v,h^{-1}v\right\rangle )^{2}\sqrt{\left\langle v,h^{-1}v\right\rangle }\\ & \leq2\alpha(1+\langle v,h^{-1}v\rangle )^{2}\sqrt{\langle v,h^{-1}v\rangle }.\nonumber \end{align} $$

Proof of Proposition 7.3.

The desired bounds follow from applying the general estimates of this section to $h_k$ and $v_k$ , plus using the uniform bound on $\alpha $ in (7.65) and the estimates

(7.85) $$ \begin{align} \left\Vert v\right\Vert ^{2}\leq C\hat{V}_{k}|k|k_{F},\quad\big\langle v_{k},h_{k}^{-1}v_{k}\big\rangle \leq C\hat{V}_{k},\quad\big\langle v_{k},h_{k}^{-\frac{3}{2}}v_{k}\big\rangle \leq C\hat{V}_{k}|k|^{3+\frac{2}{3}}(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}, \end{align} $$

which hold for all $k\in \overline {B}\left(0,2k_{F}\right)$ due to Propositions A.1, A.2, and A.3.

8 Gronwall estimates for the Bogolubov transformation

In the previous sections, we have bounded several error terms using the operators $H_{\mathrm {kin}}'$ and $\mathcal {N}_E$ . In this section, we control the propagation of these operators under the Bogolubov transformation $e^{-\mathcal {K}}$ defined in Section 5. We have the following Gronwall-type estimates.

Proposition 8.1. Let $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}|k|<\infty $ . Then for all $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ and $\left|t\right|\leq 1$ , it holds that

$$ \begin{align*} \left\langle e^{-t\mathcal{K}}\Psi, (H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) e^{-t\mathcal{K}}\Psi\right\rangle &\leq C \left\langle \Psi, ( H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \Psi\right\rangle,\\ \left\langle e^{-t\mathcal{K}}\Psi, (k_F^{-1} \mathcal{N}_E H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) e^{-t\mathcal{K}}\Psi\right\rangle &\leq C\left\langle \Psi,(k_F^{-1} \mathcal{N}_E H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \Psi\right\rangle \end{align*} $$

for a constant $C>0$ independent of $k_{F}$ .

As a preparation, let us first prove the following:

Lemma 8.2. Let $X,Y,Z$ be self-adjoint operators on a Hilbert space such that

$$ \begin{align*}X,Z>0, \quad [X,Z]=0, \quad \pm \left[\left[Y,X\right],X\right] \le Z.\end{align*} $$

Then,

$$ \begin{align*}\pm \left[\left[Y,\sqrt{X}\right],\sqrt{X}\right] \le \frac{Z}{4X}. \end{align*} $$

Proof of Lemma 8.2 .

Using (7.8), we can write

(8.1) $$ \begin{align} \left[Y,\sqrt{X}\right]=\frac{2}{\pi}\int_{0}^{\infty} \left[Y, \frac{X}{X+t^{2}}\right]\,dt= \frac{2}{\pi}\int_{0}^{\infty} \left[Y, \frac{-t^2}{X+t^{2}}\right]\,dt = \frac{2}{\pi}\int_{0}^{\infty} \frac{1}{X+t^{2}} \left[Y, X\right] \frac{1}{X+t^{2}}\, t^2 dt, \end{align} $$

and applying this identity twice, we get

(8.2) $$ \begin{align} \left[\left[Y,\sqrt{X}\right],\sqrt{X}\right] = \left(\frac{2}{\pi} \right)^2 \int_{0}^{\infty} \int_0^\infty \frac{1}{X+t^{2}} \frac{1}{X+s^{2}} \left[\left[Y, X\right],X \right] \frac{1}{X+s^{2}} \frac{1}{X+t^{2}}\, s^2 t^2 ds dt. \end{align} $$

Therefore, the assumptions $\pm \left[\left[Y,X\right],X\right] \le Z$ and $[X,Z]=0$ imply that

(8.3) $$ \begin{align} \pm \left[\left[Y,\sqrt{X}\right],\sqrt{X}\right] \le \left(\frac{2}{\pi} \right)^2 \int_{0}^{\infty} \int_0^\infty \frac{1}{X+t^{2}} \frac{1}{X+s^{2}} Z \frac{1}{X+s^{2}} \frac{1}{X+t^{2}}\, s^2 t^2 ds dt = \frac{Z}{4X}. \end{align} $$

Now we give the following:

Proof of Proposition 8.1 .

Write $\Psi _{t}=e^{t\mathcal {K}} \Psi $ for brevity. Recalling Proposition 5.4, we see that

(8.4) $$ \begin{align} \frac{d}{dt}\left\langle \Psi_{t},(H_{\operatorname{\mathrm{kin}}}^{\prime}+k_F)\Psi_{t}\right\rangle =\left\langle \Psi_{t},\left[\mathcal{K},H_{\operatorname{\mathrm{kin}}}^{\prime}\right]\Psi_{t}\right\rangle =\sum_{k\in S_{C}}\left\langle \Psi_{t},Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right)\Psi_{t}\right\rangle. \end{align} $$

The right-hand side can be bounded by using Propositions 4.10 and 7.3 as

(8.5) $$ \begin{align} & \sum_{k\in S_{C}}\left|\left\langle \Psi_{t},Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right)\Psi_{t}\right\rangle \right| \leq2\sum_{k\in S_{C}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{HS}}\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle \nonumber \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad+2\sum_{k\in S_{C}}\left\Vert \left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{HS}}\sqrt{\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle }\left\Vert \Psi_{t}\right\Vert \nonumber \\ & \leq C\left(\sum_{k\in S_{C}}\hat{V}_{k}\right)\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle +Ck_{F}^{\frac{1}{2}}\left(\sum_{k\in S_{C}}\hat{V}_{k}|k|^{\frac{1}{2}}\right)\sqrt{\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle }\left\Vert \Psi_t\right\Vert \nonumber \\ & \leq C\left\langle \Psi_{t}, (H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \Psi_{t}\right\rangle, \end{align} $$

where we also used the Cauchy–Schwarz inequality in the last step. Thus, the first estimate of Proposition 8.1 follows by Gronwall’s lemma. For the second bound of Proposition 8.1, let us denote

(8.6) $$ \begin{align} X_1 = \mathcal{N}_E +k_F \ge k_F, \quad X_2= k_F+ H_{\operatorname{\mathrm{kin}}}^{\prime} \ge k_F, \quad Y_1= [\mathcal{K}, X_2] \quad \text{ and }\quad Y_2= [\mathcal{K}, X_1]. \end{align} $$

Note that $Y_1,Y_2$ are symmetric since $X_1,X_2$ are symmetric and $\mathcal {K}$ is skew-symmetric. Moreover, since $[X_1,X_2]=0$ , $[\mathcal {K}, X_1X_2]$ is also symmetric, and we can write

(8.7) $$ \begin{align} 2[\mathcal{K}, X_1 X_2] &= 2\Big( X_1 [\mathcal{K}, X_2] + [\mathcal{K}, X_1] X_2\Big) = 2( X_1 Y_1+ Y_2X_2) \nonumber\\ &= \sum_{i=1}^2 ( X_i Y_i+ Y_iX_i) = \sum_{i=1}^2 \left( 2 \sqrt{X_i} Y_i \sqrt{X_i} + [[ Y_i, \sqrt{X_i}], \sqrt{X_i}] \right). \end{align} $$

For $i=1$ , arguing similarly to (8.4) and (8.5), we have

(8.8) $$ \begin{align} \pm Y_1 = \pm [\mathcal{K}, H_{\operatorname{\mathrm{kin}}}']= \pm \sum_{k\in S_{C}} Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right) \le C X_2, \quad \pm \sqrt{X_1} Y_1 \sqrt{X_1} \le C X_1 X_2. \end{align} $$

Here, we used $[X_1,X_2]=0$ in the last estimate. To apply Lemma 8.2, let us compute $[[Y_1, X_1], X_1]$ . Note that for every symmetric operator B on $\ell ^2(L_k^{\pm })$ , we deduce from (1.75) that

(8.9) $$ \begin{align} \left[ Q_{2}^{k}(B) ,\mathcal{N}_E\right] & = 2 \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle \left(- b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right), \nonumber\\ \left[ \left[ Q_{2}^{k}(B) ,\mathcal{N}_E\right],\mathcal{N}_E\right] &= 4 \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle \left( b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right) = 4 Q_{2}^{k}(B). \end{align} $$

Using (8.9) and (8.8), we have

(8.10) $$ \begin{align} \pm [[Y_1, X_1], X_1] = \pm 4 \sum_{k\in S_{C}} Q_{2}^{k}\left(\left\{ K_{k}^{\oplus},h_{k}^{\oplus}\right\} \right) \le C X_2, \end{align} $$

which implies by Lemma 8.2 that

(8.11) $$ \begin{align} \pm [[Y_1, \sqrt{X_1}], \sqrt{X_1}] \le C X_2 X_1^{-1}. \end{align} $$

Next, we consider the terms of $i=2$ in (8.7). Let us compute the commutator $Y_2=\left[\mathcal {K},\mathcal {N}_{E}\right]$ . By linearity, we deduce from (1.75) that $\left[b_{k}(\varphi ),\mathcal {N}_{E}\right]=b_{k}(\varphi )$ for any $\varphi \in \ell ^{2}(L_{k}^{\pm })$ , and hence from the definition of $\mathcal {K}$ in (5.2),

(8.12) $$ \begin{align} Y_2=\left[\mathcal{K},\mathcal{N}_{E}\right] =\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(e_{p})b_{k}^{\ast}\left(K_{k}^{\oplus}e_{p}\right)+b_{k}\left(K_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})\right)=\sum_{k\in S_{C}}Q_{2}^{k}\left(K_{k}^{\oplus}\right). \end{align} $$

Note that

(8.13) $$ \begin{align} K_{k}^{\oplus}= \left(\begin{array}{cc} 0 & K_{k}\\ K_{k} & 0 \end{array}\right)=\left(\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{array}\right)\left(\begin{array}{cc} K_{k} & 0\\ 0 & -K_{k} \end{array}\right)\left(\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{array}\right), \end{align} $$

and hence by Proposition 7.1, we obtain

(8.14) $$ \begin{align} \sum_{k\in S_{C}}\left\Vert K_{k}^{\oplus}\right\Vert _{\text{HS}}\leq\sum_{k\in S_{C}} \mathrm{tr} ( |K_{k}^{\oplus}| ) =2\sum_{k\in S_{C}}\text{tr}\left(\left|K_{k}\right|\right) \le C \sum_{k\in \mathbb{Z}^3} \hat V_k. \end{align} $$

Therefore, by Proposition 4.7,

(8.15) $$ \begin{align} \pm Y_2 \leq2\left(\sum_{k\in S_{C}}\left\Vert K_{k}^{\oplus}\right\Vert _{\text{HS}}\right) \left(1+\mathcal{N}_{E}\right) \le C X_1, \quad \pm \sqrt{X_2} Y_2 \sqrt{X_2} \le C X_1 X_2. \end{align} $$

Finally, consider

(8.16) $$ \begin{align} [[Y_2, X_2], X_2] = \sum_{k\in S_{C}} \left[ \left[ Q_{2}^{k}\left(K_{k}^{\oplus} \right), H_{\operatorname{\mathrm{kin}}}' \right], H_{\operatorname{\mathrm{kin}}}'\right]. \end{align} $$

For every symmetric operator B on $\ell ^2(L_k^{\pm })$ , by (1.74), we compute

(8.17) $$ \begin{align} \left[ Q_{2}^{k}(B) , H_{\operatorname{\mathrm{kin}}}' \right] & = \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle \left[ \left(b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right),H_{\operatorname{\mathrm{kin}}}' \right] \nonumber\\ &= \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle (\lambda_{\overline{k,p}} +\lambda_{ \overline{k,q}} ) \left(- b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right), \nonumber \\ \left[ \left[ Q_{2}^{k}(B) ,H_{\operatorname{\mathrm{kin}}}' \right],H_{\operatorname{\mathrm{kin}}}' \right] & = \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle (\lambda_{\overline{k,p}} +\lambda_{ \overline{k,q}} ) \left[ \left(- b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right) , H_{\operatorname{\mathrm{kin}}}' \right] \nonumber \\ &= \sum_{p,q\in L_{k}^{\pm}} \left\langle e_{p},Be_{q}\right\rangle (\lambda_{\overline{k,p}} +\lambda_{ \overline{k,q}} )^2 \left( b_{\overline{k,p}}^{\ast}b_{\overline{k,q}}^{\ast}+b_{\overline{k,q}}b_{\overline{k,p}}\right). \end{align} $$

By the Cauchy–Schwarz inequality, we can estimate

(8.18) $$ \begin{align} \pm \left[ \left[ Q_{2}^{k}(B) ,H_{\operatorname{\mathrm{kin}}}' \right],H_{\operatorname{\mathrm{kin}}}' \right] \le \sum_{p,q\in L_{k}^{\pm}} \left( \epsilon \Big( \lambda_{\overline{k,p}} +\lambda_{\overline{k,q}} \Big)^4 b_{\overline{k,p}}^{\ast} b_{\overline{k,q}}^{\ast} b_{\overline{k,q}} b_{\overline{k,p}} + \epsilon^{-1} |\left\langle e_{p},Be_{q}\right\rangle|^2 \right) \end{align} $$

for all $\epsilon>0$ . From Propositions 4.5, 4.8 and the commutation relations (1.74), (1.75), we have

(8.19) $$ \begin{align} 0&\le \sum_{p,q\in L_k^{\pm}} \lambda_{\overline{k,p}} b_{\overline{k,p}}^{\ast} b_{\overline{k,q}}^{\ast} b_{\overline{k,q}} b_{\overline{k,p}} \le \sum_{p\in L_k^{\pm}} \lambda_{\overline{k,p}} b_{\overline{k,p}}^{\ast} \mathcal{N}_E b_{\overline{k,p}} \le H_{\operatorname{\mathrm{kin}}}' \mathcal{N}_E, \nonumber\\ 0&\le \sum_{p,q\in L_k^{\pm}} \lambda_{\overline{k,q}} b_{\overline{k,p}}^{\ast} b_{\overline{k,q}}^{\ast} b_{\overline{k,q}} b_{\overline{k,p}} \le \sum_{p\in L_k^{\pm}} b_{\overline{k,p}}^{\ast} H_{\operatorname{\mathrm{kin}}}' b_{\overline{k,p}} \le H_{\operatorname{\mathrm{kin}}}' \mathcal{N}_E. \end{align} $$

Moreover, when $k\in S_{C}=\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ with $1\ge \gamma>0$ , we have

(8.20) $$ \begin{align} |\lambda_{\overline{k,p}}| \le C|k| k_F,\quad |\lambda_{\overline{k,p}}|^3 \le C |k|^3 k_F^3 \le C |k|^2 k_F^4, \quad \forall p\in L_k^{\pm}. \end{align} $$

Hence, we conclude from (8.18) that

(8.21) $$ \begin{align} \pm \left[ \left[ Q_{2}^{k}(B) ,H_{\operatorname{\mathrm{kin}}}' \right],H_{\operatorname{\mathrm{kin}}}' \right] \le C \Big( \epsilon |k|^2 k_F^4 H_{\operatorname{\mathrm{kin}}}' \mathcal{N}_E + \epsilon^{-1} \|B\|_{\mathrm{HS}}^2 \Big) \end{align} $$

for all $\epsilon>0$ . Optimizing over $\epsilon $ gives

(8.22) $$ \begin{align} \pm \left[ \left[ Q_{2}^{k}(B) ,H_{\operatorname{\mathrm{kin}}}' \right],H_{\operatorname{\mathrm{kin}}}' \right] \le C \|B\|_{\mathrm{HS}} |k| k_F \Big( H_{\operatorname{\mathrm{kin}}}' \mathcal{N}_E + k_F^2 \Big) \end{align} $$

for all symmetric operators B on $\ell ^2(L_k^{\pm })$ . Inserting this in (8.16) and using

(8.23) $$ \begin{align} \sum_{k\in S_{C}} |k| \left\Vert K_{k}^{\oplus}\right\Vert _{\text{HS}}\leq\sum_{k\in S_{C}} |k| \mathrm{tr} (|K_{k}^{\oplus}|) =2 \sum_{k\in S_{C}} |k| \text{tr}\left(\left|K_{k}\right|\right) \le C \sum_{k\in \mathbb{Z}^3} |k| \hat V_k \end{align} $$

(which is similar to (8.14)), we find that

(8.24) $$ \begin{align} \pm [[Y_1, X_2], X_2] = \pm \sum_{k\in S_{C}} \left[ \left[ Q_{2}^{k}\left(K_{k}^{\oplus} \right), H_{\operatorname{\mathrm{kin}}}' \right], H_{\operatorname{\mathrm{kin}}}'\right] \le C k_F \Big( H_{\operatorname{\mathrm{kin}}}' \mathcal{N}_E + k_F^2 \Big) \le C X_1 X_2^2. \end{align} $$

Applying Lemma 8.2, we obtain

(8.25) $$ \begin{align} \pm [[Y_1, \sqrt{X_2}], \sqrt{X_2}] \le C X_1 X_2. \end{align} $$

Putting together (8.8), (8.11), (8.15) and (8.25), we conclude from (8.7) that

(8.26) $$ \begin{align} \pm [\mathcal{K}, X_1 X_2] \le C X_1 X_2. \end{align} $$

Thus,

(8.27) $$ \begin{align} \left|\frac{d}{dt}\left\langle \Psi_{t},X_1X_2 \Psi_{t}\right\rangle \right| =\left| \left\langle \Psi_{t}, [\mathcal{K}, X_1X_2] \Psi_{t}\right\rangle \right| \le C \left\langle \Psi_{t},X_1X_2 \Psi_{t}\right\rangle. \end{align} $$

By Gronwall’s lemma, we have

(8.28) $$ \begin{align} \left\langle \Psi_{t},X_1 X_2 \Psi_{t}\right\rangle \leq C \left\langle \Psi, X_1 X_2 \Psi\right\rangle, \quad \forall |t|\le 1. \end{align} $$

This implies the desired bound since $\frac {1}{2}X_1X_2 \le \mathcal {N}_E H_{\operatorname {\mathrm {kin}}}^{\prime } + k_F H_{\operatorname {\mathrm {kin}}}^{\prime } + k_F^2 \le X_1X_2$ . Here, we used again Proposition 2.1.

9 The second Bogolubov transformation

Recall that after the conjugation by $e^{\mathcal {K}}$ , up to negligible error terms, we obtain the correlation energy and the operator

(9.1) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(E_{k}^{\oplus}-h_{k}^{\oplus}\right). \end{align} $$

In the bosonic analogy, where we informally consider $H_{\operatorname {\mathrm {kin}}}^{\prime }\sim 2\sum _{k\in \mathbb {Z}_{+}^{3}}\tilde {Q}_{1}^{k}(h_{k}^{\oplus })$ , this expression would be manifestly non-negative as $H_{\operatorname {\mathrm {kin}}}^{\prime }$ cancels the negative terms $2\sum _{k\in S_{C}}\tilde {Q}_{1}^{k}(-h_{k}^{\oplus })$ (and $E_{k}^{\oplus }>0$ as $E_{k}=e^{-K_{k}}h_{k}e^{-K_{k}}>0$ ), so this term could be neglected for the lower bound. This analogy is only formal, however. One might still hope that $E_{k}^{\oplus }-h_{k}^{\oplus }\geq 0$ since $E_{k}$ is isospectral to $\widetilde {E}_k$ and $\widetilde {E}_k\ge h_k$ , but this fails too; it can be shown that $E_{k}-h_{k}$ is indefinite. While these two ideas – the bosonic analogy and the fact that $E_k-h_{k}\geq 0$ – fail on their own, we will overcome this issue by combining them. In this section, we will carry out another unitary transformation which effectively replaces $E_k$ by $\widetilde {E}_k$ in (9.1).

Consider the unitary transformation $e^{\mathcal {J}}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ , where $\mathcal {J}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ is now of the form

(9.2) $$ \begin{align} \mathcal{J}=\sum_{k\in S_{C}}\sum_{p,q\in L_{k}^{\pm}}\left\langle e_{p},J_{k}^{\oplus}e_{q}\right\rangle b_{k}^{\ast}(e_{p})b_{k}\left(e_{q}\right)=\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p}), \end{align} $$

where $S_{C}=\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ with $1\ge \gamma>0$ and

(9.3) $$ \begin{align} J_{k}^{\oplus}=\left(\begin{array}{cc} J_{k} & 0\\ 0 & J_{k} \end{array}\right), \quad J_{k}=\log\left(U_{k}\right), \quad U_{k}=\left(h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}h_{k}^{-\frac{1}{2}}e^{K_{k}}. \end{align} $$

Here, $U_{k}:\ell ^{2}(L_{k})\rightarrow \ell ^{2}(L_{k})$ is the unitary transformation which takes $E_{k}$ to $\widetilde E_k$ , namely,

(9.4) $$ \begin{align} U_{k} E_{k}U_{k}^{\ast} =\left(h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}\left(h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}\right)^{\frac{1}{2}}=h_{k}^{\frac{1}{2}}e^{-2K_{k}}h_{k}^{\frac{1}{2}}=\widetilde E_k, \end{align} $$

and $J_k$ is the (principal) logarithm of $U_{k}$ , so that $e^{J_k}= U_k$ . Since $J_k$ is skew-symmetric, so are $J_{k}^{\oplus }$ and $\mathcal {J}$ , and hence, $e^{\mathcal {J}}$ is a unitary operator on $\mathcal {H}_{N}$ .

In the exact bosonic case, it is not difficult to see that for every skew-symmetric operator $J: V\to V$ , the unitary operator $e^{\mathcal {J}}$ with $\mathcal {J}= \text {d}\Gamma \left(J\right) = \sum _{i} a^{\ast }\left(Je_{i}\right)a(e_{i})$ is a Bogolubov transformation on $\mathcal {F}^+(V)$ which acts on a second-quantized operator as

(9.5) $$ \begin{align} e^{\mathcal{J}}\text{d}\Gamma(A)e^{-\mathcal{J}}=\text{d}\Gamma\left(e^{J}Ae^{-J}\right). \end{align} $$

Returning to the quasi-bosonic case, we will show that

(9.6) $$ \begin{align} \quad\;e^{\mathcal{J}} \left( \sum_{k\in S_{C}}\tilde{Q}_{1}^{k} ( E_{k}^{\oplus} ) \right)e^{-\mathcal{J}} \approx \sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(e^{J_{k}^{\oplus}}E_{k}^{\oplus}e^{-J_{k}^{\oplus}}\right) = \sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( \widetilde {E}_{k}^{\oplus}\right) \end{align} $$

up to error terms which are similar to the exchange terms coming from the first transformation. Moreover, although $H_{\operatorname {\mathrm {kin}}}^{\prime } \sim 2\sum _{k\in \mathbb {Z}_{+}^{3}}\tilde {Q}_{1}^{k}(h_{k}^{\oplus })$ does not hold precisely, it is valid from the point of view of commutators as explained in (1.72), which results in $H_{\operatorname {\mathrm {kin}}}^{\prime }-2\sum _{k\in \mathbb {Z}_{+}^{3}}\tilde {Q}_{1}^{k}(h_{k}^{\oplus })$ being essentially invariant under the Bogolubov transformation $e^{\mathcal {J}}$ . The overall transformation then takes the form

(9.7) $$ \begin{align} e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( E_{k}^{\oplus}-h_{k}^{\oplus}\right)\right)e^{-\mathcal{J}} \approx H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(\widetilde {E}_{k}^{\oplus}-h_{k}^{\oplus}\right), \end{align} $$

and we now have the desired non-negative operator $\widetilde {E}_{k}^{\oplus }-h_{k}^{\oplus }\ge 0$ on the right-hand side.

While the error terms in (9.7) are similar to those coming from the first transformation, they are in practice more difficult to estimate, for although we derived simple, optimal estimates for the transformation kernels $\left(K_{k}\right)_{k\in S_{C}}$ in Section 7, we cannot obtain the same for the transformation kernels $\left(J_{k}\right)_{k\in S_{C}}$ . The justification that the second transformation works as claimed will therefore take more effort than was needed for the first transformation.

9.1 Actions on the bosonizable terms

The first step of justifying (9.7) is to prove the following exact equality.

Proposition 9.1. The unitary transformation $e^{\mathcal {J}}:\mathcal {H}_{N}\rightarrow \mathcal {H}_{N}$ given in (9.2)-(9.3) satisfies

$$ \begin{align*} &e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( E_{k}^{\oplus}-h_{k}^{\oplus}\right)\right)e^{-\mathcal{J}} \\ & =H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(\widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) +2\sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{J}} {\mathcal{E}}_{3}^{k}(F_{k}^{\oplus}(t))e^{-(1-t)\mathcal{J}}dt, \end{align*} $$

where for all $k\in \mathbb {Z}_{\ast }^{3}$ and $t\in \left[0,1\right]$ , we defined the operator $F_{k}^{\oplus }(t):\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ by

$$\begin{align*}F_{k}^{\oplus} (t) = \left(\begin{array}{cc} e^{tJ_{k}} E_{k}e^{-tJ_{k}} - h_k & 0\\ 0 & e^{tJ_{k}}E_{k}e^{-tJ_{k}} - h_k \end{array}\right), \end{align*}$$

and for symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , we defined the new exchange operator

$$ \begin{align*} \mathcal{E}_{3}^{k}(A) & =2\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\mathrm{ Re}\left(b_{k}^{\ast}(Ae_{p})\varepsilon_{k,l}(e_{p};e_{q})b_{l}(J_{l}^{\oplus}e_{q})\right)\\ & =2\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\mathrm{ Re}\left(b_{k}^{\ast}(Ae_{p})\left(\delta_{p,q}c_{\overline{q-k}}c_{\overline{p-k}}^{\ast}+\delta_{\overline{p-k},\overline{q-k}}c_{q}^{\ast}c_{p}\right)b_{l}(J_{l}^{\oplus}e_{q})\right).\nonumber \end{align*} $$

We will follow the same strategy as we did when we considered the action of the quasi-bosonic Bogolubov transformation on the $Q_{1}^{k}(A)$ and $Q_{2}^{k}(B)$ terms. First, we calculate the commutator:

Proposition 9.2. For all $k\in S_{C}$ and symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , it holds that

$$\begin{align*}\left[\mathcal{J},\tilde{Q}_{1}^{k}(A)\right]=\tilde{Q}_{1}^{k}([J_{k}^{\oplus},A])+\mathcal{E}_{3}^{k}(A). \end{align*}$$

Proof. We first calculate, using the commutation relations of the excitation operators $b_{k}(\varphi )$ and $b_{k}^{\ast }(\varphi )$ , that for any $k\in S_{C}$ and $\varphi \in \ell ^{2}(L_{k}^{\pm })$ ,

(9.8) $$ \begin{align} \left[\mathcal{J},b_{k}(\varphi )\right] & =-\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(b_{l}^{\ast}(J_{l}^{\oplus}e_{q})\left[b_{k}(\varphi ),b_{l}\left(e_{q}\right)\right]+\left[b_{k}(\varphi ),b_{l}^{\ast}(J_{l}^{\oplus}e_{q})\right]b_{l}\left(e_{q}\right)\right)\nonumber \\ & =-\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\left(\delta_{k,l}\left\langle \varphi,J_{l}^{\oplus}e_{q}\right\rangle +\varepsilon_{k,l}\left(\varphi;J_{l}^{\oplus}e_{q}\right)\right)b_{l}\left(e_{q}\right)\\ & =-\sum_{q\in L_{k}^{\pm}}\left\langle \varphi,J_{k}^{\oplus}e_{q}\right\rangle b_{k}\left(e_{q}\right)-\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\varepsilon_{k,l}\left(\varphi;J_{l}^{\oplus}e_{q}\right)b_{l}\left(e_{q}\right)\nonumber \\ & =\sum_{q\in L_{k}^{\pm}}\left\langle J_{k}^{\oplus}\varphi,e_{q}\right\rangle b_{k}\left(e_{q}\right)+\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\varepsilon_{k,l}\left(\varphi;e_{q}\right)b_{l}(J_{l}^{\oplus}e_{q})\nonumber \\ & =b_{k}\left(J_{k}^{\oplus}\varphi\right)+\mathcal{E}_{k}^{\mathcal{J}}(\varphi )\nonumber \end{align} $$

for

(9.9) $$ \begin{align} \mathcal{E}_{k}^{\mathcal{J}}(\varphi )=\sum_{l\in S_{C}}\sum_{q\in L_{l}^{\pm}}\varepsilon_{k,l}\left(\varphi;e_{q}\right)b_{l}(J_{l}^{\oplus}e_{q}), \end{align} $$

where we used the skew-symmetry of $J_{k}^{\oplus }$ , anti-linearity of $\varphi \mapsto b_{k}(\varphi )$ , and Lemma 3.3. Consequently, we compute for $\tilde {Q}_{1}^{k}(A)$ that

(9.10) $$ \begin{align} &\left[\mathcal{J},\tilde{Q}_{1}^{k}(A)\right] =\sum_{p\in L_{k}^{\pm}}\left[\mathcal{J},b_{k}^{\ast}(Ae_{p})b_{k}(e_{p})\right] \nonumber \\ &\quad =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\left[\mathcal{J},b_{k}(e_{p})\right]+\left[\mathcal{J},b_{k}(Ae_{p})\right]^{\ast}b_{k}(e_{p})\right)\nonumber \\ &\quad =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\left(b_{k}\left(J_{k}^{\oplus}e_{p}\right)+\mathcal{E}_{k}^{\mathcal{J}}(e_{p})\right)+\left(b_{k}\left(J_{k}^{\oplus}Ae_{p}\right)+\mathcal{E}_{k}^{\mathcal{J}}(Ae_{p})\right)^{\ast}b_{k}(e_{p})\right)\nonumber \\ &\quad =\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})b_{k}\left(J_{k}^{\oplus}e_{p}\right)+b_{k}^{\ast}\left(J_{k}^{\oplus}Ae_{p}\right)b_{k}(e_{p})\right)\\ &\qquad +\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}(Ae_{p})\mathcal{E}_{k}^{\mathcal{J}}(e_{p})+\left(b_{k}^{\ast}(e_{p})\mathcal{E}_{k}^{\mathcal{J}}(Ae_{p})\right)^{\ast}\right)\nonumber \\ &\quad =\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}\left(\left(J_{k}^{\oplus}A-AJ_{k}^{\oplus}\right)e_{p}\right)b_{k}(e_{p})+2\sum_{l\in S_{C}}\sum_{p\in L_{k}^{\pm}}\sum_{q\in L_{l}^{\pm}}\mathrm{ Re}\left(b_{k}^{\ast}(Ae_{p})\varepsilon_{k,l}(e_{p};e_{q})b_{l}(J_{l}^{\oplus}e_{q})\right)\nonumber \\ &\quad =\tilde{Q}_{1}^{k}([J_{k}^{\oplus},A])+\mathcal{E}_{3}^{k}(A).\nonumber \end{align} $$

To derive an expression for $e^{\mathcal {J}}\tilde {Q}_{1}^{k}(A)e^{-\mathcal {J}}$ , we will use the Baker-Campbell-Hausdorff formula

(9.11) $$ \begin{align} \exp\left(\mathcal{C}_{J_{k}^{\oplus}}\right)(A)=\sum_{m=0}^{\infty}\frac{1}{m!}\mathcal{C}_{J_{k}^{\oplus}}^{m}(A)=e^{J_{k}^{\oplus}}Ae^{-J_{k}^{\oplus}},\quad \text{with } \mathcal{C}_{J_{k}^{\oplus}}(A)=\left[J_{k}^{\oplus},A\right]. \end{align} $$

Imitating the proof of Proposition 5.3, we deduce the following:

Proposition 9.3. For all $k\in S_{C}$ and symmetric $A:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ , it holds that

$$\begin{align*}e^{\mathcal{J}}\tilde{Q}_{1}^{k}(A)e^{-\mathcal{J}}=\tilde{Q}_{1}^{k}\left(e^{J_{k}^{\oplus}}Ae^{-J_{k}^{\oplus}}\right)+\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(e^{(1-t)J_{k}^{\oplus}}Ae^{-(1-t)J_{k}^{\oplus}}\right)e^{-t\mathcal{J}}dt, \end{align*}$$

the integrals being Riemann integrals of bounded operators.

Proof. We claim that for any $n\in \mathbb {N}$ , it holds that

(9.12) $$ \begin{align} e^{\mathcal{J}}\tilde{Q}_{1}^{k}(A)e^{-\mathcal{J}} & =\tilde{Q}_{1}^{k}\left(\sum_{m=0}^{n-1}\frac{1}{m!}\mathcal{C}_{J_{k}^{\oplus}}^{m}(A)\right)+\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(\sum_{m=0}^{n-1}\frac{1}{m!}\mathcal{C}_{(1-t)J_{k}^{\oplus}}^{m}(A)\right)e^{-t\mathcal{J}}dt \nonumber\\ &\quad +\frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{J}}\tilde{Q}_{1}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{J}}(1-t)^{n-1}dt. \end{align} $$

We proceed by induction. For $n=1$ , we have by the fundamental theorem of calculus and Proposition 9.2 that

(9.13) $$ \begin{align} e^{\mathcal{J}}\tilde{Q}_{1}^{k}(A)e^{-\mathcal{J}} & =\tilde{Q}_{1}^{k}(A)+\int_{0}^{1}e^{t\mathcal{J}}\left[\mathcal{J},\tilde{Q}_{1}^{k}(A)\right]e^{-t\mathcal{J}}\,dt \nonumber\\ & =\tilde{Q}_{1}^{k}(A)+\int_{0}^{1}e^{t\mathcal{J}}\tilde{Q}_{1}^{k}([J_{k}^{\oplus},A])e^{-t\mathcal{J}}\,dt+\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}(A)e^{-t\mathcal{J}}\,dt, \end{align} $$

which is the claim. For the inductive step, we assume that case n holds and integrate the last term of equation (9.12) by parts:

(9.14) $$ \begin{align} & \frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{J}}\tilde{Q}_{1}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{J}}(1-t)^{n-1}dt\nonumber \\ &\quad =\frac{1}{(n-1)!}\left[e^{t\mathcal{J}}\tilde{Q}_{1}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{J}}\left(-\frac{(1-t)^{n}}{n}\right)\right]_{0}^{1}\nonumber \\ &\qquad -\frac{1}{(n-1)!}\int_{0}^{1}e^{t\mathcal{J}}\left[\mathcal{J},\tilde{Q}_{1}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)\right]e^{-t\mathcal{J}}\left(-\frac{(1-t)^{n}}{n}\right)dt\\ &\quad =\frac{1}{n!}\tilde{Q}_{1}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)+\frac{1}{n!}\int_{0}^{1}e^{t\mathcal{J}}\left(\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right]\right)+\mathcal{E}_{3}^{k}\left(\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)\right)e^{-t\mathcal{J}}(1-t)^{n}dt\nonumber \\ &\quad =\tilde{Q}_{1}^{k}\left(\frac{1}{n!}\mathcal{C}_{J_{k}^{\oplus}}^{n}(A)\right)+\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(\frac{1}{n!}\mathcal{C}_{(1-t)J_{k}^{\oplus}}^{n}(A)\right)e^{-t\mathcal{J}}\,dt \nonumber\\ &\qquad +\frac{1}{n!}\int_{0}^{1}e^{t\mathcal{J}} \tilde{Q}_1^k \left( \mathcal{C}_{J_{k}^{\oplus}}^{n+1}(A) \right)e^{-t\mathcal{J}}(1-t)^{n}dt.\nonumber \end{align} $$

Insertion of this identity into equation (9.12) yields the statement for case $n+1$ , so our claim (9.12) holds. We can now take $n\rightarrow \infty $ and appeal to equation (9.11) to get the claim.

Proposition 9.2 also allows us to describe the action of $e^{\mathcal {J}}$ on $H_{\operatorname {\mathrm {kin}}}^{\prime }$ :

Proposition 9.4. It holds that

$$\begin{align*}e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)\right)e^{-\mathcal{J}}=H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)-2\sum_{k\in S_{C}}\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(h_{k}^{\oplus}\right)e^{-t\mathcal{J}}dt. \end{align*}$$

Proof. By the fundemental theorem of calculus and the fact that $\partial _t (e^{tA}B e^{-tA})= e^{tA}[A,B]e^{-tA}$ , the left side is equal to

$$\begin{align*} H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right) +\int_{0}^{1}e^{t\mathcal{J}}\left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)\right]e^{-t\mathcal{J}}dt. \end{align*}$$

Recalling (1.74), we may compute using Lemma 3.3 that

(9.15) $$ \begin{align} \left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}\right] & =-\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left[H_{\operatorname{\mathrm{kin}}}^{\prime},b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})\right]\nonumber \\ &\quad -2\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(-b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}\left(h_{k}^{\oplus}e_{p}\right)+b_{k}^{\ast}\left(h_{k}^{\oplus}J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})\right) \nonumber\\ & =2\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}b_{k}^{\ast}\left(\left(J_{k}^{\oplus}h_{k}^{\oplus}-h_{k}^{\oplus}J_{k}^{\oplus}\right)e_{p}\right)b_{k}(e_{p})=2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right). \end{align} $$

Combining with Proposition 9.2, we have that

(9.16) $$ \begin{align} \left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)\right]=\left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}\right]-2\sum_{k\in S_{C}}\left[\mathcal{J},\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)\right]=-2\sum_{k\in S_{C}}\mathcal{E}_{3}^{k}\left(h_{k}^{\oplus}\right), \end{align} $$

which implies the claim.

We can now conclude:

Proof of Proposition 9.1 .

By the Propositions 9.3 and 9.4, we see that

(9.17) $$ \begin{align} & e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}- 2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( h_{k}^{\oplus}\right) + 2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( E_{k}^{\oplus}\right)\right)e^{-\mathcal{J}}\nonumber \\ &\quad =H_{\operatorname{\mathrm{kin}}}^{\prime}-2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(h_{k}^{\oplus}\right)-2\sum_{k\in S_{C}}\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(h_{k}^{\oplus}\right)e^{-t\mathcal{J}}\,dt \nonumber\\ &\qquad +2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(e^{J_{k}^{\oplus}} E_{k}^{\oplus}e^{-J_{k}^{\oplus}}\right)+2\int_{0}^{1}e^{t\mathcal{J}}\mathcal{E}_{3}^{k}\left(e^{(1-t)J_{k}^{\oplus}} E_{k}^{\oplus}e^{-(1-t)J_{k}^{\oplus}}\right)e^{-t\mathcal{J}}dt\nonumber \\ &\quad =H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(e^{J_{k}^{\oplus}} E_{k}^{\oplus}e^{-J_{k}^{\oplus}}-h_{k}^{\oplus}\right)+2\int_{0}^{1}e^{(1-t)\mathcal{J}}\mathcal{E}_{3}^{k}\left(e^{tJ_{k}^{\oplus}} E_{k}^{\oplus}e^{-tJ_{k}^{\oplus}}-h_{k}^{\oplus}\right)e^{-(1-t)\mathcal{J}}dt, \end{align} $$

where we also reparametrized the integral. From the choice of $J_k^{\oplus }$ in (9.3), we have

(9.18) $$ \begin{align} e^{tJ_{k}^{\oplus}} E_{k}^{\oplus}e^{-tJ_{k}^{\oplus}} - h_k^{\oplus} = \left(\begin{array}{cc} e^{tJ_{k}} E_k e^{-tJ_k} - h_k & 0\\ 0 & e^{tJ_{k}} E_k e^{-tJ_k} -h_k \end{array}\right) ={E}_k^{\oplus}(t) \end{align} $$

for all $t\in [0,1]$ . Moreover, using $e^{J_k}=U_k$ and (9.4), we get $e^{J_{k}^{\oplus }} E_{k}^{\oplus }e^{-tJ_{k}^{\oplus }}=\widetilde {E}_k^{\oplus }$ .

9.2 Estimates for the exchange terms

Now we estimate the new exchange term $\mathcal {E}_3$ in Proposition 9.1. We have the following:

Proposition 9.5. For all $k\in \mathbb {Z}_{+}^{3}$ , symmetric $E:\ell ^{2}(L_{k}^{\pm })\rightarrow \ell ^{2}(L_{k}^{\pm })$ and $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ , it holds that

$$\begin{align*}\left|\left\langle \Psi,\mathcal{E}_{3}^{k}(E)\Psi\right\rangle \right|\leq C\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Ee_{p}\right\Vert \right)\left(\sum_{l\in S_{C}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}J_{l}^{\oplus}\right\Vert _{\operatorname{\mathrm{HS}}}\right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle } \end{align*}$$

for a constant $C>0$ independent of all quantities.

Proof of Proposition 9.5 .

We can follow the analysis in Section 6. In particular, the same reduction in Section 6.1 applies to $\mathcal {E}_{3}^{k}(E)$ , but in this case, it is significantly simpler. By definition, up to taking adjoints, every term of $\mathcal {E}_{3}^{k}(E)$ immediately reduces to the schematic form

(9.19) $$ \begin{align} \sum_{l\in S_{C}}\sum_{p\in S}b_{k}^{\ast}\left(Ee_{p_{1}}\right)\tilde{c}_{p_{3}}^{\ast}\tilde{c}_{p_{4}}b_{l}\left(J_{l}^{\oplus}e_{p_{2}}\right), \end{align} $$

and recalling that commutators of the forms $\left[\tilde {c}_{p},b_{k}(\varphi )\right]$ and $\left[\tilde {c}_{p}^{\ast },b_{k}^{\ast }(\varphi )\right]$ also vanish, we may normal-order this schematic form without introducing additional terms. Controlling $\mathcal {E}_{3}^{k}(E)$ thus reduces entirely to the estimation of the single schematic form

(9.20) $$ \begin{align} \sum_{l\in S_{C}}\sum_{p\in S}\tilde{c}_{p_{3}}^{\ast}b_{k}^{\ast}\left(Ee_{p_{1}}\right)b_{l}\left(J_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}. \end{align} $$

We estimate the schematic form of equation (9.20) using Proposition 4.4, Lemma 6.6 and the Cauchy-Schwarz inequality:

(9.21) $$ \begin{align} & \sum_{l\in S_{C}}\sum_{p\in S}\left|\left\langle \Psi,\tilde{c}_{p_{3}}^{\ast}b_{k}^{\ast}\left(Ee_{p_{1}}\right)b_{l}\left(J_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\rangle \right|\leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert b_{k}\left(Ee_{p_{1}}\right)\tilde{c}_{p_{3}}\Psi\right\Vert \left\Vert b_{l}\left(J_{l}^{\oplus}e_{p_{2}}\right)\tilde{c}_{p_{4}}\Psi\right\Vert \nonumber \\ &\quad \leq\sum_{l\in S_{C}}\sum_{p\in S}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Ee_{p_{1}}\right\Vert \left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}J_{l}^{\oplus}e_{p_{2}}\right\Vert \sqrt{\left\langle \tilde{c}_{p_{3}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{3}}\Psi\right\rangle \left\langle \tilde{c}_{p_{4}}\Psi,H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{4}}\Psi\right\rangle } \nonumber\\ & \quad\leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Ee_{p}\right\Vert \right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }\sum_{l\in S_{C}}\sqrt{\sum_{p\in S}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}J_{l}^{\oplus}e_{p_{2}}\right\Vert ^{2}}\sqrt{\sum_{p\in S}\left\langle \Psi,\tilde{c}_{p_{4}}^{\ast}H_{\text{kin}}^{\prime\left(\pm1\right)}\tilde{c}_{p_{4}}\Psi\right\rangle }\nonumber \\ &\quad \leq\left(\max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}Ee_{p_{1}}\right\Vert \right)\left(\sum_{l\in S_{C}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}J_{l}^{\oplus}\right\Vert _{\text{HS}}\right)\sqrt{\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle }. \end{align} $$

9.3 One-body operator estimates

In this subsection, we derive estimates on the one-body quantities

(9.22) $$ \begin{align} \max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}E_{k} (t )e_{p}\right\Vert ,\quad\left\Vert h_{k}^{-\frac{1}{2}}J_{k}\right\Vert_{\text{HS}},\quad \quad\left\Vert h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}}\right\Vert_{\text{HS}}, \quad \mathrm{tr} ( h_k^{-1/2} (\widetilde{E}_k - h_k) h_k^{-1/2}). \end{align} $$

The first two quantities arise from the analysis of the exchange terms in the previous subsection, while the third quantity will be needed in order to derive Gronwall-type estimates for the kinetic operator. The last one is useful to remove the cutoff $S_C$ on the right-hand side of (9.7) at the end. The estimates we will establish are the following:

Proposition 9.6. Assume $\sum _{k\in \mathbb {Z}^3_{*}}\hat V_k |k|<\infty $ . Then for all $k\in \mathbb {Z}^3_{*}$ , we have

$$ \begin{align*} \mathrm{tr} \Big( h_k^{-1/2} (\widetilde{E}_k - h_k) h_k^{-1/2}\Big) \le C \hat V_k. \end{align*} $$

Moreover, if $k\in \overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{\ast }^{3}$ , $0<\gamma <\frac {1}{47}$ and $t\in \left[0,1\right]$ , it holds that

$$ \begin{align*} \max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}E_{k} (t ) e_{p}\right\Vert & \leq Ck_{F}^{-\frac{1}{2}}\left(\hat{V}_{k}+\hat{V}_{k}^{3}|k|^{6}\log\left(k_{F}\right)\right),\\ \left\Vert h_{k}^{-\frac{1}{2}}J_{k}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\hat{V}_{k},\\ \left\Vert h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}} & \leq C\hat{V}_{k}. \end{align*} $$

Here, the constant $C>0$ is independent of k and $k_{F}$ .

Proposition 9.6 is the main source of the technical restriction $\gamma <\frac 1 {47}$ which comes from the use of the first bound in Proposition A.3 (we need $\gamma <\frac {4+3\beta }{8-3\beta }$ with $\beta =-\frac 5 4$ ).

As in Section 7, in order to simplify the notation, let $h:V\rightarrow V$ denote a self-adjoint operator acting on an n-dimensional Hilbert space V, let $(x_{i})_{i=1}^{n}$ denote an eigenbasis for h with eigenvalues $\left(\lambda _{i}\right)_{i=1}^{n}$ , and let $v\in V$ be any vector such that $\left\langle v,x_{i}\right\rangle \geq 0$ for all $1\leq i\leq n$ . As before, we take

(9.23) $$ \begin{align} K=-\frac{1}{2}\log\left(h^{-\frac{1}{2}}\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{2}}h^{-\frac{1}{2}}\right). \end{align} $$

We will establish general estimates for the operators

(9.24) $$ \begin{align} U=\left(h^{\frac{1}{2}}e^{-2K}h^{\frac{1}{2}}\right)^{\frac{1}{2}}h^{-\frac{1}{2}}e^{K},\quad J = \log(U),\quad E (t )=e^{tJ}e^{-K}he^{-K}e^{-tJ} -h \end{align} $$

and then at the end insert the explicit choice (7.2) to get the desired estimates.

Unlike the case in Section 7, we will now also take V to be a complex Hilbert space. This is not a strictly necessary assumption, but it allows us to streamline the presentation significantly, as it implies that the unitary operator U is diagonalizable and so lets us describe the operators $J=\log (U)$ and $e^{tJ}$ solely in terms of eigenvectors of U.

The main difficulty of the proof of Proposition 9.6 is that we cannot extend the argument leading to matrix element estimates for $e^{-2K}-1$ and $1-e^{2K}$ in Section 7 to handle the operators J and $e^{tJ}$ . Instead, we will utilize a technique which effectively lets us replace relevant quantities of J by these of $U-1$ , by exploiting the diagonalizability of U.

We start with the easy part of Proposition 9.6.

Proposition 9.7. With $\widetilde {E} = \left(h^{2}+2P_{h^{\frac {1}{2}}v}\right)^{\frac {1}{2}}$ , we have

$$ \begin{align*}\mathrm{tr} \Big(h^{-1/2} (\widetilde{E} - h) h^{-1/2}\Big) \le \langle v, h^{-1}v\rangle.\end{align*} $$

Proof. Using (7.21) for $\widetilde {E} = \left(h^{2}+2P_{h^{\frac {1}{2}}v}\right)^{\frac {1}{2}}$ , we can write

(9.25) $$ \begin{align} h^{-1/2} (\widetilde{E} - h) h^{-1/2} = \frac{4}{\pi}\int_{0}^{\infty}\frac{t^{2}}{1+2\left\langle v,h\left(h^{2}+t^{2}\right)^{-1}v\right\rangle }P_{\left(h^{2}+t^{2}\right)^{-1}v}\,dt \le \frac{4}{\pi}\int_{0}^{\infty} P_{\left(h^{2}+t^{2}\right)^{-1}v} t^2\,dt. \end{align} $$

Taking the trace and using (7.8), we complete the proof.

Estimates for U

Let us consider the unitary operator $U:V\rightarrow V$ defined by

(9.26) $$ \begin{align} U=(h^{\frac{1}{2}}e^{-2K}h^{\frac{1}{2}})^{\frac{1}{2}}h^{-\frac{1}{2}}e^{K}=(h^{2}+2P_{h^{\frac{1}{2}}v})^{\frac{1}{4}}h^{-\frac{1}{2}}e^{K}. \end{align} $$

First, the analysis of $(h^{2}+2P_{h^{\frac {1}{2}}v})^{\frac {1}{2}}$ in Section 7 can be extended to $(h^{2}+2P_{h^{\frac {1}{2}}v})^{\frac {1}{4}}$ . We have the following:

Proposition 9.8. For all $1\leq i,j\leq n$ , it holds that

$$\begin{align*}\left|\left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)x_{j}\right\rangle \right|\leq\frac{2\sqrt{\lambda_{i}\lambda_{j}}}{\sqrt{\lambda_{i}}+\sqrt{\lambda_{j}}}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}. \end{align*}$$

Note that by using the integral identity

(9.27) $$ \begin{align} A^{\frac{1}{4}}=\frac{2\sqrt{2}}{\pi}\int_{0}^{\infty}\left(1-\frac{t^{4}}{A+t^{4}}\right)dt \end{align} $$

for every self-adjoint non-negative operator A instead of (7.8), we obtain the following analogue of Proposition 7.5:

Proposition 9.9. Let $\left(H,\left\langle \cdot ,\cdot \right\rangle \right)$ be a Hilbert space and let $A:H\rightarrow H$ be a positive self-adjoint operator. Then for any $x\in H$ and $g\in \mathbb {R}$ such that $A+gP_{x}>0$ , it holds that

$$\begin{align*}\left(A+gP_{x}\right)^{\frac{1}{4}}=A^{\frac{1}{4}}+\frac{2\sqrt{2}g}{\pi}\int_{0}^{\infty}\frac{t^{4}}{1+g\left\langle v,\left(A+t^{4}\right)^{-1}v\right\rangle }P_{\left(A+t^{4}\right)^{-1}v}\,dt. \end{align*}$$

Proof of Proposition 9.8 .

Applying Proposition 9.9 with $A=h^{2}$ , $x=h^{\frac {1}{2}}v$ and $g=2$ , we find

(9.28) $$ \begin{align} \left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}} & =\left(h^{2}\right)^{\frac{1}{4}}+\frac{4\sqrt{2}}{\pi}\int_{0}^{\infty}\frac{t^{4}}{1+2\left\langle h^{\frac{1}{2}}v,\left(h^{2}+t^{4}\right)^{-1}h^{\frac{1}{2}}v\right\rangle }P_{\left(h^{2}+t^{4}\right)^{-1}h^{\frac{1}{2}}v}\,dt\\ & =h^{\frac{1}{2}}+\frac{4\sqrt{2}}{\pi}\int_{0}^{\infty}\frac{t^{4}}{1+2\left\langle v,h\left(h^{2}+t^{4}\right)^{-1}v\right\rangle }P_{h^{\frac{1}{2}}\left(h^{2}+t^{4}\right)^{-1}v}\,dt,\nonumber \end{align} $$

and so we can estimate that

(9.29) $$ \begin{align} 0 & \le \left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)x_{j}\right\rangle \nonumber \\ & =\frac{4\sqrt{2}}{\pi}\int_{0}^{\infty}\frac{t^{4}}{1+2\left\langle v,h\left(h^{2}+t^{4}\right)^{-1}v\right\rangle }\left\langle x_{i},P_{h^{\frac{1}{2}}\left(h^{2}+t^{4}\right)^{-1}v}x_{j}\right\rangle dt\nonumber \\ & =\frac{4\sqrt{2}}{\pi}\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \int_{0}^{\infty}\frac{t^{4}}{1+2\left\langle v,h\left(h^{2}+t^{4}\right)^{-1}v\right\rangle }\frac{\sqrt{\lambda_{i}}}{\lambda_{i}^{2}+t^{4}}\frac{\sqrt{\lambda_{j}}}{\lambda_{j}^{2}+t^{4}}\,dt\\ & \leq\frac{4\sqrt{2}}{\pi}\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \int_{0}^{\infty}\frac{\sqrt{\lambda_{i}}}{\lambda_{i}^{2}+t^{4}}\frac{\sqrt{\lambda_{j}}}{\lambda_{j}^{2}+t^{4}}t^{4}\,dt=\frac{2\sqrt{\lambda_{i}\lambda_{j}}}{\sqrt{\lambda_{i}}+\sqrt{\lambda_{j}}}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}},\nonumber \end{align} $$

where we also applied the integral identity

(9.30) $$ \begin{align} \int_{0}^{\infty}\frac{\sqrt{a}}{a^{2}+t^{4}}\frac{\sqrt{b}}{b^{2}+t^{4}}t^{4}\,dt=\frac{\pi}{2\sqrt{2}}\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\frac{1}{a+b},\quad a,b>0. \end{align} $$

We may then conclude the following:

Proposition 9.10. For all $1\leq i,j\leq n$ , it holds that

$$\begin{align*}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|,\,\left|\left\langle x_{i},\left(U^{\ast}-1\right)x_{j}\right\rangle \right|\leq3\left(1+\left\langle v,h^{-1}v\right\rangle \right)\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}. \end{align*}$$

Proof. As $\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|=\left|\left\langle x_{j},\left(U^{\ast }-1\right)x_{i}\right\rangle \right|$ and the claimed estimate is symmetric with respect to i and j, it suffices to consider $U-1$ . We write

(9.31) $$ \begin{align} U-1 & =\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}h^{-\frac{1}{2}}e^{K}-1=\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}e^{K}+e^{K}-1\\ & =e^{K}-1+\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}+\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}\left(e^{K}-1\right)\nonumber \end{align} $$

and estimate each term separately. The first is directly covered by Proposition 7.10, with

(9.32) $$ \begin{align} \left|\left\langle x_{i},\left(e^{K}-1\right)x_{j}\right\rangle \right|\leq\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}. \end{align} $$

For the second term, we can by Proposition 9.8 estimate that

(9.33) $$ \begin{align} \left|\left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}x_{j}\right\rangle \right| & =\frac{1}{\sqrt{\lambda_{j}}}\left|\left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)x_{j}\right\rangle \right|\\ & \leq\frac{1}{\sqrt{\lambda_{j}}}\frac{2\sqrt{\lambda_{i}\lambda_{j}}}{\sqrt{\lambda_{i}}+\sqrt{\lambda_{j}}}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\leq2\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}.\nonumber \end{align} $$

For the final term, we carry out an orthonormal expansion and apply the previous two estimates to see that

(9.34) $$ \begin{align} & \left|\left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}\left(e^{K}-1\right)x_{j}\right\rangle \right|\nonumber \\ &\quad \leq\sum_{k=1}^{n}\left|\left\langle x_{i},\left(\left(h^{2}+2P_{h^{\frac{1}{2}}v}\right)^{\frac{1}{4}}-h^{\frac{1}{2}}\right)h^{-\frac{1}{2}}x_{k}\right\rangle \right|\left|\left\langle x_{k},\left(e^{K}-1\right)x_{j}\right\rangle \right| \nonumber\\ & \quad\leq2\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}=2\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\sum_{k=1}^{n}\frac{\lambda_{i}+\lambda_{j}}{\left(\lambda_{i}+\lambda_{k}\right)\left(\lambda_{k}+\lambda_{j}\right)}\left|\left\langle x_{k},v\right\rangle \right|^{2}\nonumber \\ &\quad \leq2\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}=2\left\langle v,h^{-1}v\right\rangle \frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}} , \end{align} $$

where we also applied the elementary inequality

(9.35) $$ \begin{align} \frac{a+b}{\left(a+c\right)\left(c+b\right)}=\frac{a+b}{c(a+b)+ ab + c^2} < \frac{1}{c},\quad \forall a,b,c>0. \end{align} $$

Combining the estimates now yields the claim.

Estimates for J

Recall that we defined $J:V\rightarrow V$ to be the principal logarithm of U. Since U is a unitary operator on the finite-dimensional complex Hilbert space V, by the spectral theorem it is diagonalizable – that is, there exists an orthonormal basis $(w_{j})_{j=1}^{n}$ for V of eigenstates of U with eigenvalues $\left(e^{i\theta _{j}}\right)_{j=1}^{n}$ , $\left(\theta _{j}\right)_{j=1}^{n}\subset \left(-\pi ,\pi \right]$ , (i.e., $Uw_{j}=e^{i\theta _{j}}w_{j}$ for all $1\leq j\leq n$ ). Thus, J can be explicitly written as

(9.36) $$ \begin{align} Jw_{j}=i\theta_{j}w_{j},\quad1\leq j\leq n. \end{align} $$

To estimate the quantity $\left\Vert h^{-\frac {1}{2}}J\right\Vert _{\text {HS}}$ , we will apply the following:

Proposition 9.11. It holds that

$$\begin{align*}JJ^{\ast}\leq\frac{\pi^{2}}{4}\left(U-1\right)^{\ast}\left(U-1\right). \end{align*}$$

Proof. We note the elementary inequality

(9.37) $$ \begin{align} \left|x\right|\leq\frac{\pi}{2}\sqrt{2\left(1-\cos (x )\right)}=\frac{\pi}{2}\left|e^{ix}-1\right|,\quad x\in\left[-\pi,\pi\right], \end{align} $$

which can be deduced from the fact that $x\mapsto \left|e^{ix}-1\right|$ is an even function and concave on $x\in \left[0,\pi \right]$ . As the eigenbasis $(w_{j})_{j=1}^{n}$ obeys

(9.38) $$ \begin{align} Uw_{j}=e^{i\theta_{j}}w_{j},\quad U^{\ast}w_{j}=e^{-i\theta_{j}}w_{j},\quad Jw_{j}=i\theta_{j}w_{j},\quad J^{\ast}w_{j}=-i\theta_{j}w_{j}, \end{align} $$

we can for any $w\in V$ perform an orthonormal expansion in terms of $(w_{j})_{j=1}^{n}$ to see that

(9.39) $$ \begin{align} \left\langle w,JJ^{\ast}w\right\rangle & =\left\Vert J^{\ast}w\right\Vert ^{2}=\sum_{j=1}^{n}\left|\theta_{j}\right|^{2}\left|\left\langle w_{j},w\right\rangle \right|^{2}\leq\sum_{j=1}^{n}\left(\frac{\pi}{2}\left|e^{i\theta_{j}}-1\right|\right)^{2}\left|\left\langle w_{j},w\right\rangle \right|^{2} \nonumber\\ & =\frac{\pi^{2}}{4}\sum_{j=1}^{n}\left|\left\langle \left(U^{\ast}-1\right)w_{j},w\right\rangle \right|^{2}=\frac{\pi^{2}}{4}\left\Vert \left(U-1\right)w\right\Vert ^{2}=\frac{\pi^{2}}{4}\left\langle w,\left(U-1\right)^{\ast}\left(U-1\right)w\right\rangle , \end{align} $$

which is the claim.

Corollary 9.12. There exists a universal constant $C>0$ such that

$$\begin{align*}\left\Vert h^{-\frac{1}{2}}J\right\Vert _{\operatorname{\mathrm{HS}}}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\langle v,h^{-\frac{3}{2}}v\right\rangle. \end{align*}$$

Proof. By cyclicity of the trace and the estimate of the previous proposition, we have that

(9.40) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}J\right\Vert _{\text{HS}}^{2} & =\text{tr}\left(J^{\ast}h^{-1}J\right)=\text{tr}\left(h^{-\frac{1}{2}}JJ^{\ast}h^{-\frac{1}{2}}\right) \nonumber\\ & \leq\frac{\pi^{2}}{4}\text{tr}\left(h^{-\frac{1}{2}}\left(U-1\right)^{\ast}\left(U-1\right)h^{-\frac{1}{2}}\right)=\frac{\pi^{2}}{4}\left\Vert \left(U-1\right)h^{-\frac{1}{2}}\right\Vert _{\text{HS}}^{2}, \end{align} $$

and by the matrix element estimate of Proposition 9.10,

(9.41) $$ \begin{align} & \quad\;\left\Vert \left(U-1\right)h^{-\frac{1}{2}}\right\Vert _{\text{HS}}^{2}=\sum_{i,j=1}^{n}\left|\left\langle x_{i},\left(U-1\right)h^{-\frac{1}{2}}x_{j}\right\rangle \right|^{2}=\sum_{i,j=1}^{n}\frac{1}{\lambda_{j}}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|^{2} \nonumber\\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i,j=1}^{n}\frac{1}{\lambda_{j}}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i,j=1}^{n}\frac{1}{\lambda_{j}}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}^{\frac{3}{4}}\lambda_{j}^{\frac{1}{4}}}\right|^{2}\nonumber \\ & =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{j}^{\frac{3}{2}}}\right)^{2}=C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-\frac{3}{2}}v\right\rangle ^{2}, \end{align} $$

which gives the claim.

Next, consider $\Vert h^{-\frac {1}{2}}[J,h]h^{-\frac {1}{2}}\Vert _{\text {HS}}$ . By the triangle inequality, it suffices to bound $\Vert h^{-\frac {1}{2}}Jh^{\frac {1}{2}}\Vert _{\text {HS}}$ . Unlike $\Vert h^{-\frac {1}{2}}J\Vert _{\text {HS}}$ , this is more involved as the presence of factors of h on both sides of J prevents us from combining J and $J^{\ast }$ in $\Vert h^{-\frac {1}{2}}Jh^{\frac {1}{2}}\Vert _{\text {HS}}^{2}=\text {tr}(J^{\ast }h^{-1}Jh)$ , and so we need to proceed differently. First, we note the following elementary estimate:

Lemma 9.13. There exists a constant $C>0$ such that

$$\begin{align*}\left|i\theta-\frac{1}{2}\left(e^{i\theta}-e^{-i\theta}\right)\right|\leq C\left|e^{i\theta}-1\right|^{3},\quad\theta\in\left[-\pi,\pi\right]. \end{align*}$$

Proof. The left-hand side is $|\theta -\sin (\theta )|=O(|\theta |^3)$ , while $ |\theta | \ge \left|e^{i\theta }-1\right| \ge C^{-1}\theta $ .

Proposition 9.14. There exists a universal constant $C>0$ such that

$$\begin{align*}\left\Vert h^{-\frac{1}{2}}\left[J,h\right]h^{-\frac{1}{2}}\right\Vert _{\operatorname{\mathrm{HS}}}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{3}\left(\left\langle v,h^{-1}v\right\rangle +\left\langle v,h^{-\frac{1}{2}}v\right\rangle \left\langle v,h^{-\frac{5}{4}}h\right\rangle ^{2}\right). \end{align*}$$

Proof. It suffices to bound $\Vert h^{-\frac {1}{2}}Jh^{\frac {1}{2}}\Vert _{\text {HS}}$ . By writing

(9.42) $$ \begin{align} J=\frac{1}{2}\left(U-1\right)+\frac{1}{2}\left(1-U^{\ast}\right) + \widetilde{J}, \quad \widetilde{J} = J-\frac{1}{2}\left(U-U^{\ast}\right), \end{align} $$

we see by the triangle inequality that

(9.43) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}Jh^{\frac{1}{2}}\right\Vert _{\text{HS}}\leq\frac{1}{2}\left\Vert h^{-\frac{1}{2}}\left(U-1\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}+\frac{1}{2}\left\Vert h^{-\frac{1}{2}}\left(1-U^{\ast}\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}+\left\Vert h^{-\frac{1}{2}} \widetilde{J} h^{\frac{1}{2}}\right\Vert _{\text{HS}}. \end{align} $$

By Proposition 9.10, we have

(9.44) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}\left(U-1\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2} &=\sum_{i,j=1}^{n}\left|\left\langle x_{i},h^{-\frac{1}{2}}\left(U-1\right)h^{\frac{1}{2}}x_{j}\right\rangle \right|^{2}=\sum_{i,j=1}^{n}\frac{\lambda_{j}}{\lambda_{i}}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i,j=1}^{n}\frac{\lambda_{j}}{\lambda_{i}}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2}\nonumber \\ &\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\right)^{2} \nonumber\\ & =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle ^{2} \end{align} $$

and likewise for $\Vert h^{-\frac {1}{2}}(1-U^{\ast })h^{\frac {1}{2}}\Vert _{\text {HS}}^{2}$ . For $h^{-\frac {1}{2}}\widetilde {J} h^{\frac {1}{2}}$ , we instead apply Lemma 9.13 and the Cauchy-Schwarz inequality to see that for any $1\leq i,j\leq n$ ,

(9.45) $$ \begin{align} \left|\left\langle x_{i},h^{-\frac{1}{2}} \widetilde{J} h^{\frac{1}{2}}x_{j}\right\rangle \right|^2 & =\left|\sum_{k=1}^{n}\left(i\theta_{k}-\frac{1}{2}\left(e^{i\theta_{k}}-e^{-i\theta_{k}}\right)\right)\left\langle h^{-\frac{1}{2}}x_{i},w_{k}\right\rangle \left\langle w_{k},h^{\frac{1}{2}}x_{j}\right\rangle \right|^2 \nonumber \\ & \leq C \left( \sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{3}\left|\left\langle h^{-\frac{1}{2}}x_{i},w_{k}\right\rangle \right|\left|\left\langle w_{k},h^{\frac{1}{2}}x_{j}\right\rangle \right| \right)^2 \nonumber\\ & \leq C \left(\sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{4}\left|\left\langle w_{k},h^{-\frac{1}{2}}x_{i}\right\rangle \right|^{2}\right) \left( \sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{2}\left|\left\langle w_{k},h^{\frac{1}{2}}x_{j}\right\rangle \right|^{2} \right)\nonumber \\ & =C \left( \sum_{k=1}^{n}\left|\left\langle \left(U^{\ast}-1\right)^{2}w_{k},h^{-\frac{1}{2}}x_{i}\right\rangle \right|^{2} \right) \left( \sum_{k=1}^{n}\left|\left\langle \left(U^{\ast}-1\right)w_{k},h^{\frac{1}{2}}x_{j}\right\rangle \right|^{2} \right) \nonumber \\ & =C\left\Vert \left(U-1\right)^{2}h^{-\frac{1}{2}}x_{i}\right\Vert^2 \left\Vert \left(U-1\right)h^{\frac{1}{2}}x_{j}\right\Vert^2. \end{align} $$

Summing over $i,j$ , we obtain

(9.46) $$ \begin{align} \left\Vert h^{-\frac{1}{2}} \widetilde{J} h^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2} \le C\left\Vert \left(U-1\right)^{2}h^{-\frac{1}{2}}\right\Vert _{\text{HS}}^{2}\left\Vert \left(U-1\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2}. \end{align} $$

We can now again apply Proposition 9.10 to estimate that

(9.47) $$ \begin{align} \left\Vert \left(U-1\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}^{2} &=\sum_{i,j=1}^{n}\left|\left\langle x_{i},\left(U-1\right)h^{\frac{1}{2}}x_{j}\right\rangle \right|^{2}=\sum_{i,j=1}^{n}\lambda_{j}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i,j=1}^{n}\lambda_{j}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2} \nonumber\\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i,j=1}^{n}\lambda_{j}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}^{\frac{1}{4}}\lambda_{j}^{\frac{3}{4}}}\right|^{2}=C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-\frac{1}{2}}v\right\rangle ^{2} \end{align} $$

and

(9.48) $$ \begin{align} \left\Vert \left(U-1\right)^{2}h^{-\frac{1}{2}}\right\Vert _{\text{HS}}^{2} &=\sum_{i,j=1}^{n}\left|\left\langle x_{i},\left(U-1\right)^{2}h^{-\frac{1}{2}}x_{j}\right\rangle \right|^{2}\nonumber \\ & =\sum_{i,j=1}^{n}\frac{1}{\lambda_{j}}\left|\sum_{k=1}^{n}\left\langle x_{i},\left(U-1\right)x_{k}\right\rangle \left\langle x_{k},\left(U-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\frac{1}{\lambda_{j}}\left(\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right)^{2} \nonumber\\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\frac{\left|\left\langle x_{j},v\right\rangle \right|^{2}}{\lambda_{j}}\left(\sum_{k=1}^{n}\frac{\left\langle v,x_{k}\right\rangle }{\lambda_{i}^{\frac{5}{8}}\lambda_{k}^{\frac{3}{8}}}\frac{\left\langle x_{k},v\right\rangle }{\lambda_{k}^{\frac{7}{8}}\lambda_{j}^{\frac{1}{8}}}\right)^{2}\nonumber \\ & =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left(\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{5}{4}}}\right)^{4}=C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{4} \end{align} $$

so

(9.49) $$ \begin{align} \big\Vert h^{-\frac{1}{2}} \widetilde{J} h^{\frac{1}{2}}\big\Vert _{\text{HS}}\leq\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{3}\big\langle v,h^{-\frac{1}{2}}v\big\rangle \big\langle v,h^{-\frac{5}{4}}v\big\rangle ^{2}. \end{align} $$

Combining the estimates yields the claim.

Remark 9.1 (Remarks on the estimation technique).

As we will use the same approach to obtain estimates on $E (t )$ , let us consider the technique of the proof in detail. The idea is that, as we have a good estimate for the matrix elements of $U-1$ and $U^{\ast }-1$ , we should attempt to express our operator solely in terms of these. The first step is therefore to decompose J as in (9.42). The error term $\widetilde {J}=J-\frac {1}{2}\left(U-U^{\ast }\right) $ cannot be simplified further in terms of U but by orthonormal expansion and Lemma 9.13, we can nonetheless estimate it solely in terms of $U-1$ , despite being unable to apply an operator inequality, as we did for $\| h^{-\frac {1}{2}}J\|_{\text {HS}}$ , to ‘substitute’ $U-1$ for J directly. The utility of the estimate (9.46) is thus that it allows us to replace the unknown error operator with factors of $U-1$ , which we can estimate well. The downside to this is that it simultanously ‘decouples’ the $h^{-\frac {1}{2}}$ and $h^{\frac {1}{2}}$ factors, which prevents us from exploiting the cancellation between these.

This decoupling is also the reason why it is important that in (9.46) we distribute two factors of $U-1$ to $h^{-\frac {1}{2}}$ rather than only one. One can by the same argument estimate that

(9.50) $$ \begin{align} \big\Vert h^{-\frac{1}{2}} \widetilde{J} h^{\frac{1}{2}}\big\Vert _{\text{HS}} & \leq C\big\Vert \big(U-1\big)h^{-\frac{1}{2}}\big\Vert _{\text{HS}}\big\Vert \big(U-1\big)^{2}h^{\frac{1}{2}}\big\Vert _{\text{HS}} \nonumber\\ & \leq C\big(1+\big\langle v,h^{-3}v\big\rangle \big)^{3}\big\langle v,h^{-\frac{3}{4}}v\big\rangle ^{2}\big\langle v,h^{-\frac{3}{2}}v\big\rangle , \end{align} $$

but in Proposition A.3, we only have the good estimates $\big\langle v_{k},h_{k}^{\alpha }v_{k}\big\rangle \sim Ck_{F}^{1+\alpha }$ for $\alpha>-\frac {4}{3}$ , which makes (9.50) a worse estimate due to the $\big\langle v,h^{-\frac {3}{2}}v\big\rangle $ factor. There is therefore a limit to how low the exponent $\alpha $ can be without affecting our estimates, and so it is advantageous to distribute the factors of $U-1$ such that the overall minimal exponent is not too small.

Estimation of $E (t )$

We now estimate $\max _{j}\Vert h^{-\frac {1}{2}}E (t )x_{j}\Vert $ using the technique outlined above. First, we decompose

(9.51) $$ \begin{align} E (t ) =e^{tJ}e^{-K}he^{-K}e^{-tJ}-h=(e^{tJ}he^{-tJ}-h)+e^{tJ}(e^{-K}he^{-K}-h)e^{-tJ}=:E_{1} (t )+E_{2} (t ) \end{align} $$

and using the algebraic identity

(9.52) $$ \begin{align} ABC=B+\left(A-1\right)B+B\left(C-1\right)+\left(A-1\right)B\left(C-1\right) \end{align} $$

with $A=e^{tJ}$ , $B=h$ and $C=e^{-tJ}$ further decompose $E_{1} (t )$ as

(9.53) $$ \begin{align} E_{1} (t )=e^{tJ}he^{-tJ}-h & =\big(\big(e^{tJ}-1\big)h+h\big(e^{-tJ}-1\big)\big)+\big(e^{tJ}-1\big)h\big(e^{-tJ}-1\big) \nonumber\\ & =:E_{1,1} (t )+E_{1,2} (t ). \end{align} $$

Defining $E_{0}=E\big(0\big)=e^{-K}he^{-K}-h$ , we likewise decompose $E_{2} (t )$ according to

(9.54) $$ \begin{align} E_{2} (t )=e^{tJ}E_{0}e^{-tJ} & =E_{0}+\big(\big(e^{tJ}-1\big)E_{0}+E_{0}\big(e^{-tJ}-1\big)\big)+\big(e^{tJ}-1\big)E_{0}\big(e^{-tJ}-1\big) \nonumber\\ & =:E_{0}+E_{2,1} (t )+E_{2,2} (t ). \end{align} $$

The $E_{1,1} (t )$ , $E_{1,2} (t )$ and $E_{2,1} (t )$ , $E_{2,2} (t )$ terms differ only in replacing the operator h by $E_{0}$ . We can therefore estimate these terms similarly, provided we have an estimate on $E_{0}$ . This is given by the following:

Proposition 9.15. For all $1\leq i,j\leq n$ , it holds that

$$\begin{align*}\big|\big\langle x_{i},E_{0}x_{j}\big\rangle \big|=\big|\big\langle x_{i},\big(e^{-K}he^{-K}-h\big)x_{j}\big\rangle \big|\leq\big(1+\big\langle v,h^{-1}v\big\rangle \big)\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle. \end{align*}$$

Consequently,

$$\begin{align*}\max_{1\leq j\leq n}\big\Vert h^{-\frac{1}{2}}E_{0}x_{j}\big\Vert \leq\alpha\big(1+\big\langle v,h^{-1}v\big\rangle \big)\sqrt{\big\langle v,h^{-1}v\big\rangle }, \end{align*}$$

where $\alpha =\max _{1\leq j\leq n}\big\langle v,x_{j}\big\rangle $ .

Proof. Using the identity of equation (9.52) with $A=e^{-K}=C$ and $B=h$ , we have that

(9.55) $$ \begin{align} e^{-K}he^{-K}-h=\big\{ h,e^{-K}-1\big\} +\big(e^{-K}-1\big)h\big(e^{-K}-1\big). \end{align} $$

Hence,

(9.56) $$ \begin{align} \big\langle x_{i},\big(e^{-K}he^{-K}-h\big)x_{j}\big\rangle =\big(\lambda_{i}+\lambda_{j}\big)\big\langle x_{i},\big(e^{-K}-1\big)x_{j}\big\rangle +\big\langle x_{i},\big(e^{-K}-1\big)h\big(e^{-K}-1\big)x_{j}\big\rangle. \end{align} $$

We can apply Proposition 7.10 to estimate the first term of this equation as

(9.57) $$ \begin{align} \big|\big(\lambda_{i}+\lambda_{j}\big)\big\langle x_{i},\big(e^{-K}-1\big)x_{j}\big\rangle \big|\leq\big(\lambda_{i}+\lambda_{j}\big)\frac{\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle }{\lambda_{i}+\lambda_{j}}=\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle \end{align} $$

and the second term as

(9.58) $$ \begin{align} \big|\big\langle x_{i},\big(e^{-K}-1\big)h\big(e^{-K}-1\big)x_{j}\big\rangle \big| & =\big|\sum_{k=1}^{n}\lambda_{k}\big\langle x_{i},\big(e^{-K}-1\big)x_{k}\big\rangle \big\langle x_{k},\big(e^{-K}-1\big)x_{j}\big\rangle \big|\nonumber \\ & \leq\sum_{k=1}^{n}\lambda_{k}\frac{\big\langle x_{i},v\big\rangle \big\langle v,x_{k}\big\rangle }{\lambda_{i}+\lambda_{k}}\frac{\big\langle x_{k},v\big\rangle \big\langle v,x_{j}\big\rangle }{\lambda_{k}+\lambda_{j}} \nonumber\\ & \leq\big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle \sum_{k=1}^{n}\frac{\big|\big\langle x_{k},v\big\rangle \big|^{2}}{\lambda_{k}}=\big\langle v,h^{-1}v\big\rangle \big\langle x_{i},v\big\rangle \big\langle v,x_{j}\big\rangle , \end{align} $$

which implies the first claim. Consequently,

(9.59) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{0}x_{j}\right\Vert ^{2} &=\sum_{i=1}^{n}\left|\left\langle x_{i},h^{-\frac{1}{2}}\left(e^{-K}he^{-K}-h\right)x_{j}\right\rangle \right|^{2}=\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\left\langle x_{i},\left(e^{-K}he^{-K}-h\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle \right|^{2} \leq\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle. \end{align} $$

Now it remains to consider the operators $e^{tJ}-1$ and $e^{-tJ}-1=\left(e^{tJ}-1\right)^{\ast }$ . To implement the above estimation technique, from the following analogue of Lemma 9.13,

(9.60) $$ \begin{align} \left|\big(e^{it\theta}-1\big)-t\big(e^{i\theta}-1\big)+\frac{t(1-t)}{2}\big(e^{i\theta}+e^{-i\theta}-2\big)\right|\leq C\left|e^{i\theta}-1\right|^{3},\quad t\in\left[0,1\right],\,\theta\in\left[-\pi,\pi\right], \end{align} $$

we are motivated in approximating $e^{tJ}-1$ by

(9.61) $$ \begin{align} F_{t}=t\left(U-1\right)-\frac{t(1-t)}{2}\left(U+U^{\ast}-2\right),\quad t\in\left[0,1\right], \end{align} $$

with the error term being cubic with respect to $U-1$ . We then have the following bounds for $F_{t}$ and the associated error terms:

Proposition 9.16. For any $T:V\rightarrow V$ , $x\in V$ , $m\in \left\{ 1,2\right\} $ and $t\in \left[0,1\right]$ , it holds that

$$\begin{align*}\left\Vert T\big(e^{tJ}-1-F_{t}\big)x\right\Vert ,\,\left\Vert T\big(e^{-tJ}-1-F_{t}^{\ast}\big)x\right\Vert \leq C\left\Vert T\big(U-1\big)^{m}\right\Vert _{\operatorname{\mathrm{HS}}}\left\Vert \big(U-1\big)^{3-m}x\right\Vert \end{align*}$$

and for all $1\leq i,j\leq n$ , $t\in \left[0,1\right]$ ,

$$\begin{align*}\left|\left\langle x_{i},F_{t}x_{j}\right\rangle \right|,\,\left|\left\langle x_{i},F_{t}^{\ast}x_{j}\right\rangle \right|\leq C\big(1+\left\langle v,h^{-1}v\right\rangle \big)\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}} \end{align*}$$

for a constant $C>0$ independent of all quantities.

Proof. Recall that $(w_{j})_{j=1}^{n}$ is an orthonormal eigenbasis of J, namely, $e^{tJ}w_{j}=e^{it\theta _{j}} w_j$ for all $1\leq j\leq n$ . Using (9.60) and the Cauchy-Schwarz inequality, we have that

(9.62) $$ \begin{align} & \left\Vert T\left(e^{tJ}-1-F_{t}\right)x\right\Vert ^{2}=\sum_{j=1}^{n}\left|\left\langle w_{j},T\left(e^{tJ}-1-F_{t}\right)x\right\rangle \right|^{2}\nonumber \\ &\quad =\sum_{j=1}^{n}\left|\sum_{k=1}^{n}\left(\left(e^{it\theta_{k}}-1\right)-t\left(e^{i\theta_{k}}-1\right)+\frac{t(1-t)}{2}\left(e^{i\theta_{k}}+e^{-i\theta_{k}}-2\right)\right)\left\langle w_{j},Tw_{k}\right\rangle \left\langle w_{k},x\right\rangle \right|^{2}\nonumber \\ &\quad \leq C\sum_{j=1}^{n}\left(\sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{3}\left|\left\langle w_{j},Tw_{k}\right\rangle \right|\left|\left\langle w_{k},x\right\rangle \right|\right)^{2}\nonumber\\ &\quad \leq C\sum_{j=1}^{n}\left(\sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{2m}\left|\left\langle w_{j},Tw_{k}\right\rangle \right|^{2}\right)\left(\sum_{k=1}^{n}\left|e^{i\theta_{k}}-1\right|^{2\left(3-m\right)}\left|\left\langle w_{k},x\right\rangle \right|^{2}\right)\nonumber \\ &\quad =C\left(\sum_{j,k=1}^{n}\left|\left\langle w_{j},T\left(U-1\right)^{m}w_{k}\right\rangle \right|^{2}\right)\left(\sum_{k=1}^{n}\left|\left\langle \left(U^{\ast}-1\right)^{3-m}w_{k},x\right\rangle \right|^{2}\right)\nonumber \\ &\quad =C\left\Vert T\left(U-1\right)^{m}\right\Vert _{\text{HS}}^{2}\left\Vert \left(U-1\right)^{3-m}x\right\Vert ^{2}, \end{align} $$

the same estimate holding also for $\left\Vert T\left(e^{-tJ}-1-F_{t}^{\ast }\right)x\right\Vert $ . For the matrix element estimate of $F_{t}$ , we have by Proposition 9.10 that

(9.63) $$ \begin{align} \left|\left\langle x_{i},F_{t}x_{j}\right\rangle \right| & =\left|\left\langle x_{i},\left(\frac{t\left(1+t\right)}{2}\left(U-1\right)-\frac{t(1-t)}{2}\left(U^{\ast}-1\right)\right)x_{j}\right\rangle \right|\nonumber \\ & \leq\frac{t\left(1+t\right)}{2}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|+\frac{t(1-t)}{2}\left|\left\langle x_{i},\left(U^{\ast}-1\right)x_{j}\right\rangle \right| \nonumber\\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}} \end{align} $$

as we only consider $t\in \left[0,1\right]$ , and likewise for $\left|\left\langle x_{i},F_{t}^{\ast }x_{j}\right\rangle \right|$ .

Estimation of $E_{1} (t )$

We are now ready to estimate $E_{1} (t )=E_{1,1} (t )+E_{1,2} (t )$ , starting with $E_{1,1} (t )=\left(e^{tJ}-1\right)h+h\left(e^{-tJ}-1\right)$ . Recall that $(x_i)_i$ are an eigenbasis of h with $\langle x_i, v\rangle \ge 0$ for all $1\le i\le n$ .

Proposition 9.17. For all $t\in \left[0,1\right]$ , it holds that

$$ \begin{align*} \max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}E_{1,1} (t )x_j\right\Vert & \leq C\alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)\sqrt{\left\langle v,h^{-1}v\right\rangle }\\ &\quad + C\alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)^{3}\left(\left\Vert v\right\Vert \left\langle v,h^{-\frac{5}{4}}h\right\rangle ^{2}+\left\langle v,h^{-\frac{1}{2}}v\right\rangle \left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}\right), \end{align*} $$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ and $C>0$ is a constant independent of all quantities.

Proof. We write

(9.64) $$ \begin{align} E_{1,1} (t )=F_{t}h+hF_{t}^{\ast}+(e^{tJ}-1-F_{t})h+h(e^{-tJ}-1-F_{t}^{\ast}) \end{align} $$

so that for any $1\leq j\leq n$ , we can estimate by Proposition 9.16

(9.65) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{1,1} (t )x_{j}\right\Vert & \leq\left\Vert h^{-\frac{1}{2}}F_{t}hx_{j}\right\Vert +\left\Vert h^{\frac{1}{2}}F_{t}^{\ast}x_{j}\right\Vert +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)hx_{j}\right\Vert \nonumber\\ &\quad +C\left\Vert h^{\frac{1}{2}}\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert . \end{align} $$

We consider each term above for the following. By Proposition 9.16, we see that independently of $1\leq j\leq n$ ,

(9.66) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}F_{t}hx_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\frac{\lambda_{j}^{2}}{\lambda_{i}}\left|\left\langle x_{i},F_{t}x_{j}\right\rangle \right|^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\frac{\lambda_{j}^{2}}{\lambda_{i}}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2}\nonumber \\ & \leq C\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle ,\nonumber \\ \left\Vert h^{\frac{1}{2}}F^*_{t}x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\lambda_{i}\left|\left\langle x_{i},F^*_{t}x_{j}\right\rangle \right|^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\lambda_{i}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2} \nonumber\\ & \leq C\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle . \end{align} $$

For the remaining terms of equation (9.65), we recall that we already estimated $\big\Vert h^{-\frac {1}{2}}\big(U-1\big)^{2}\big\Vert _{\text {HS}}$ and $\big\Vert h^{\frac {1}{2}}\big(U-1\big)\big\Vert _{\text {HS}}$ in the equations (9.47) and (9.48) to be

(9.67) $$ \begin{align} \left\Vert h^{\frac{1}{2}}\left(U-1\right)\right\Vert _{\text{HS}} & =\left\Vert \left(U-1\right)h^{\frac{1}{2}}\right\Vert _{\text{HS}}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)\left\langle v,h^{-\frac{1}{2}}v\right\rangle , \\ \left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}} & =\left\Vert \left(U-1\right)^{2}h^{-\frac{1}{2}}\right\Vert _{\text{HS}}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2},\nonumber \end{align} $$

the equalities holding by normality of U. The only unknown quantities are thus $\left\Vert \left(U-1\right)hx_{j}\right\Vert $ and $\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert $ , which we estimate using Proposition 9.10 as

(9.68) $$ \begin{align} \left\Vert \left(U-1\right)hx_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\lambda_{j}^{2}\left|\left\langle x_{i},\left(U-1\right)x_{j}\right\rangle \right|^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\lambda_{j}^{2}\left|\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{i}+\lambda_{j}}\right|^{2} \nonumber\\ & \leq C\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\Vert v\right\Vert ^{2}, \end{align} $$

(9.69) $$ \begin{align} \left\Vert \left(U-1\right)^{2}x_{j}\right\Vert ^{2} & =\sum_{i=1}^{n}\left|\sum_{k=1}^{n}\left\langle x_{i},\left(U-1\right)x_{k}\right\rangle \left\langle x_{k},\left(U-1\right)x_{j}\right\rangle \right|^{2}\nonumber \\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2} \qquad\qquad\qquad\qquad\nonumber\\ & \leq C\left|\left\langle v,x_{j}\right\rangle \right|^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{2}{3}}\lambda_{k}^{\frac{4}{3}}}\right)^{2}\nonumber \\ & \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{3}. \end{align} $$

Thus,

(9.70) $$ \begin{align} \big\Vert h^{-\frac{1}{2}}\big(U-1\big)^{2}\big\Vert _{\text{HS}}\big\Vert \big(U-1\big)hx_{j}\big\Vert & \leq C\alpha\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{3}\big\Vert v\big\Vert \big\langle v,h^{-\frac{5}{4}}h\big\rangle ^{2} \nonumber\\ \big\Vert h^{\frac{1}{2}}\big(U-1\big)\big\Vert _{\text{HS}}\big\Vert \big(U-1\big)^{2}x_{j}\big\Vert & \leq C\alpha\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{3}\big\langle v,h^{-\frac{1}{2}}v\big\rangle \big\langle v,h^{-\frac{4}{3}}v\big\rangle ^{\frac{3}{2}}, \end{align} $$

which, upon combination with the estimates of equation (9.66), imply the claim.

Proposition 9.18. For all $t\in \big[0,1\big]$ , it holds that

$$ \begin{align*} & \left(C\alpha\right)^{-1}\max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}E_{1,2} (t )x_{j}\right\Vert \\ &\quad \leq\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle ^{\frac{3}{2}}+\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-\frac{1}{2}}v\right\rangle ^{2}\left\langle v,h^{-\frac{5}{4}}h\right\rangle ^{2}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}\\ &\qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left(\sqrt{\left\langle v,h^{-1}v\right\rangle }\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}+\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\right), \end{align*} $$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ and $C>0$ is a constant independent of all quantities.

Proof. We write $E_{1,2} (t )=\left(e^{tJ}-1\right)h\left(e^{-tJ}-1\right)$ as

(9.71) $$ \begin{align} E_{1,2} (t )= F_{t}hF_{t}^{\ast}+F_{t}h\big(e^{-tJ}-1-F_{t}^{\ast}\big)+\big(e^{tJ}-1-F_{t}\big)hF_{t}^{\ast}+\big(e^{tJ}-1-F_{t}\big)h\big(e^{-tJ}-1-F_{t}^{\ast}\big) \end{align} $$

and see by Proposition 9.16 that

(9.72) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{1,2} (t )x_{j}\right\Vert & \leq\left\Vert h^{-\frac{1}{2}}F_{t}hF_{t}^{\ast}x_{j}\right\Vert +C\left\Vert h^{-\frac{1}{2}}F_{t}h\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert \nonumber \\ &\quad +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)hF_{t}^{\ast}x_{j}\right\Vert \\ &\quad +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)h\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert .\nonumber \end{align} $$

We estimate by Propositions 9.10 and 9.16 that

(9.73) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}F_{t}hF_{t}^{\ast}x_{j}\right\Vert ^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\lambda_{k}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2} \nonumber\\ &\quad \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\right)^{2}=C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-1}v\right\rangle ^{3}, \end{align} $$

and

(9.74) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}F_{t}h\left(U-1\right)\right\Vert _{\text{HS}}^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\lambda_{k}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2}\nonumber \\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left|\left\langle x_{j},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}^{\frac{2}{3}}\lambda_{j}^{\frac{1}{3}}}\right)^{2} \nonumber\\ &\quad =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-1}v\right\rangle \left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}, \end{align} $$

and

(9.75) $$ \begin{align} & \left\Vert \left(U-1\right)hF_{t}^{\ast}x_{j}\right\Vert ^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\sum_{k=1}^{n}\lambda_{k}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2} \nonumber\\ &\quad \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{1}{3}}\lambda_{k}^{\frac{2}{3}}}\right)^{2}=C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}, \end{align} $$

and

(9.76) $$ \begin{align} & \left\Vert \left(U-1\right)h\left(U-1\right)\right\Vert _{\text{HS}}^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\left|\sum_{k=1}^{n}\lambda_{k}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2} \nonumber\\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left|\left\langle x_{j},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{1}{4}}\lambda_{j}^{\frac{1}{4}}\lambda_{k}^{\frac{1}{2}}}\right)^{2} \nonumber\\ &\quad =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{1}{2}}v\right\rangle ^{4}. \end{align} $$

Combining these with the estimates of the equations (9.67) and (9.69) yields

(9.77) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}F_{t}h\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert & \leq C\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sqrt{\left\langle v,h^{-1}v\right\rangle }\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}},\nonumber \\ \left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)hF_{t}^{\ast}x_{j}\right\Vert & \leq C\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2} \end{align} $$

and

(9.78) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)h\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert \nonumber\\ &\quad \leq C\alpha\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-\frac{1}{2}}v\right\rangle ^{2}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}, \end{align} $$

which imply the claim.

Estimation of $E_{2} (t )$

We now repeat the same steps for $E_{2} (t )=E_{0}+E_{2,1} (t )+E_{2,2} (t )$ where

(9.79) $$ \begin{align} E_{2,1} (t )=F_{t}E_{0}+E_{0}F_{t}^{\ast}+\left(e^{tJ}-1-F_{t}\right)E_{0}+E_{0}\left(e^{-tJ}-1-F_{t}^{\ast}\right). \end{align} $$

Proposition 9.19. For all $t\in \left[0,1\right]$ , it holds that

$$ \begin{align*} \max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}E_{2,1} (t )x\right\Vert &\leq C\alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle ^{\frac{3}{2}} \\ &\quad + C\alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\\ &\quad +C\alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sqrt{\left\langle v,h^{-1}v\right\rangle }\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}, \end{align*} $$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ and $C>0$ is a constant independent of all quantities.

Proof. By Proposition 9.16, we can estimate that

(9.80) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{2,1} (t )x_{j}\right\Vert & \leq\left\Vert h^{-\frac{1}{2}}F_{t}E_{0}x_{j}\right\Vert +\left\Vert h^{-\frac{1}{2}}E_{0}F_{t}^{\ast}x_{j}\right\Vert +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)E_{0}x_{j}\right\Vert \nonumber \\ &\quad +C\left\Vert h^{-\frac{1}{2}}E_{0}\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert. \end{align} $$

Then let us consider each term separately. By the Propositions 9.10, 9.15, 9.16, we have that

(9.81) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}F_{t}E_{0}x_{j}\right\Vert ^{2} &\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle \right|^{2} \nonumber\\ & \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\right)^{2}\nonumber\\ & =C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-1}v\right\rangle ^{3}, \end{align} $$

and

(9.82) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}E_{0}F_{t}x_{j}\right\Vert ^{2}\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle \frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2} \nonumber\\ & \quad\leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\right)^{2}=C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-1}v\right\rangle ^{3}, \end{align} $$

and

(9.83) $$ \begin{align} \left\Vert \left(U-1\right)E_{0}x_{j}\right\Vert ^{2} &\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\sum_{k=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle \right|^{2} \nonumber\\ & \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{1}{3}}\lambda_{k}^{\frac{2}{3}}}\right)^{2}\nonumber\\ &=C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}, \end{align} $$

and

(9.84) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{0}\left(U-1\right)\right\Vert _{\text{HS}}^{2} &\leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k=1}^{n}\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle \frac{\left\langle x_{k},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{k}+\lambda_{j}}\right|^{2}\nonumber \\ & \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4}\sum_{i,j=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left|\left\langle x_{j},v\right\rangle \right|^{2}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}^{\frac{2}{3}}\lambda_{j}^{\frac{1}{3}}}\right)^{2}\nonumber\\ & =C(1+\langle v,h^{-1}v\rangle )^{4}\langle v,h^{-1}v\rangle \langle v,h^{-\frac{2}{3}}v\rangle ^{3}. \end{align} $$

Combining these with our prior estimates that

(9.85) $$ \begin{align} \big\Vert h^{-\frac{1}{2}}\big(U-1\big)^{2}\big\Vert _{\text{HS}} & \leq C\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{2}\big\langle v,h^{-\frac{5}{4}}v\big\rangle ^{2}, \nonumber\\ \big\Vert \big(U-1\big)^{2}x_{j}\big\Vert _{\text{HS}} & \leq C\alpha\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{2}\big\langle v,h^{-\frac{4}{3}}v\big\rangle ^{\frac{3}{2}}, \end{align} $$

we obtain the claim.

Proposition 9.20. For all $t\in \big[0,1\big]$ , it holds that

$$ \begin{align*} & \big(C\alpha\big)^{-1}\max_{1\leq j\leq n}\big\Vert h^{-\frac{1}{2}}E_{2,2} (t )x_{j}\big\Vert \\ &\quad \leq\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{3}\big\langle v,h^{-1}v\big\rangle ^{\frac{5}{2}}+\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{7}\big\langle v,h^{-\frac{2}{3}}\big\rangle ^{3}\big\langle v,h^{-\frac{5}{4}}v\big\rangle ^{2}\big\langle v,h^{-\frac{4}{3}}v\big\rangle ^{\frac{3}{2}}\\ &\qquad +\big(1+\big\langle v,h^{-1}v\big\rangle \big)^{5} \big\langle v,h^{-\frac{2}{3}}v\big\rangle ^{\frac{3}{2}} \big(\big\langle v,h^{-1}v\big\rangle ^{\frac{3}{2}}\big\langle v,h^{-\frac{4}{3}}v\big\rangle ^{\frac{3}{2}}+\big\langle v,h^{-1}v\big\rangle \big\langle v,h^{-\frac{5}{4}}v\big\rangle ^{2}\big), \end{align*} $$

where $\alpha =\max _{1\leq j\leq n}\big\langle v,x_{j}\big\rangle $ and $C>0$ is a constant independent of all quantities.

Proof. We decompose $E_{2,2} (t )=\big(e^{tJ}-1\big)E_{0}\big(e^{-tJ}-1\big)$ as

(9.86) $$ \begin{align} F_{t}E_{0}F_{t}^{\ast}+F_{t}E_{0}\big(e^{-tJ}-1-F_{t}^{\ast}\big)+\big(e^{tJ}-1-F_{t}\big)E_{0}F_{t}^{\ast}+\big(e^{tJ}-1-F_{t}\big)E_{0}\big(e^{-tJ}-1-F_{t}^{\ast}\big) \end{align} $$

and estimate by Proposition 9.16 that

(9.87) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}E_{2,2} (t )x_{j}\right\Vert & \leq\left\Vert h^{-\frac{1}{2}}F_{t}E_{0}F_{t}^{\ast}x_{j}\right\Vert +C\left\Vert h^{-\frac{1}{2}}F_{t}E_{0}\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert \nonumber \\ &\quad +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)E_{0}F_{t}^{\ast}x_{j}\right\Vert \nonumber\\ &\quad +C\left\Vert h^{-\frac{1}{2}}\left(U-1\right)^{2}\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)E_{0}\left(U-1\right)\right\Vert _{\text{HS}}\left\Vert \left(U-1\right)^{2}x_{j}\right\Vert . \end{align} $$

We estimate as in the previous proposition that

(9.88) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}F_{t}E_{0}F_{t}^{\ast}x_{j}\right\Vert ^{2} \nonumber\\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k,l=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{l}\right\rangle \frac{\left\langle x_{l},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{l}+\lambda_{j}}\right|^{2}\nonumber \\ &\quad \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left(\sum_{k=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\frac{\left|\left\langle x_{l},v\right\rangle \right|^{2}}{\lambda_{l}}\right)^{2} \nonumber\\ &\quad =C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-1}v\right\rangle ^{5}, \end{align} $$

and

(9.89) $$ \begin{align} & \left\Vert h^{-\frac{1}{2}}F_{t}E_{0}\left(U-1\right)\right\Vert _{\text{HS}}^{2} \nonumber\\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i,j=1}^{n}\frac{1}{\lambda_{i}}\left|\sum_{k,l=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{l}\right\rangle \frac{\left\langle x_{l},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{l}+\lambda_{j}}\right|^{2}\nonumber \\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i,j=1}^{n}\frac{\left|\left\langle x_{i},v\right\rangle \right|^{2}}{\lambda_{i}}\left|\left\langle x_{j},v\right\rangle \right|^{2}\left(\sum_{k,l=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{k}}\frac{\left|\left\langle x_{l},v\right\rangle \right|^{2}}{\lambda_{l}^{\frac{2}{3}}\lambda_{j}^{\frac{1}{3}}}\right)^{2}\nonumber\\ &\quad =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-1}v\right\rangle ^{3}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}, \end{align} $$

and

(9.90) $$ \begin{align} & \left\Vert \left(U-1\right)E_{0}F_{t}^{\ast}x_{j}\right\Vert ^{2} \nonumber\\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i=1}^{n}\left|\sum_{k,l=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{l}\right\rangle \frac{\left\langle x_{l},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{l}+\lambda_{j}}\right|^{2}\nonumber \\ &\quad \leq C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left(\sum_{k,l=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{1}{3}}\lambda_{k}^{\frac{2}{3}}}\frac{\left|\left\langle x_{l},v\right\rangle \right|^{2}}{\lambda_{l}}\right)^{2} \nonumber\\ &\quad =C\alpha^{2}\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-1}v\right\rangle ^{2}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}, \end{align} $$

and finally that

(9.91) $$ \begin{align} & \left\Vert \left(U-1\right)E_{0}\left(U-1\right)\right\Vert _{\text{HS}}^{2} \nonumber\\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i,j=1}^{n}\left|\sum_{k,l=1}^{n}\frac{\left\langle x_{i},v\right\rangle \left\langle v,x_{k}\right\rangle }{\lambda_{i}+\lambda_{k}}\left\langle x_{k},v\right\rangle \left\langle v,x_{l}\right\rangle \frac{\left\langle x_{l},v\right\rangle \left\langle v,x_{j}\right\rangle }{\lambda_{l}+\lambda_{j}}\right|^{2}\nonumber \\ &\quad \leq C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\sum_{i,j=1}^{n}\left|\left\langle x_{i},v\right\rangle \right|^{2}\left|\left\langle x_{j},v\right\rangle \right|^{2}\left|\sum_{k,l=1}^{n}\frac{\left|\left\langle x_{k},v\right\rangle \right|^{2}}{\lambda_{i}^{\frac{1}{3}}\lambda_{k}^{\frac{2}{3}}}\frac{\left|\left\langle x_{l},v\right\rangle \right|^{2}}{\lambda_{l}^{\frac{2}{3}}\lambda_{j}^{\frac{1}{3}}}\right|^{2} \nonumber\\ &\quad =C\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-\frac{2}{3}}\right\rangle ^{6}. \end{align} $$

Combining these bounds with equation (9.85) yields the claim.

Combining the estimates from Proposition 9.17 through 9.20 and the last bound of Proposition 9.15, we obtain

(9.92) $$ \begin{align} & \left(C\alpha\right)^{-1}\max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}E (t )x_{j}\right\Vert \nonumber\\ &\quad \leq\left(1+\left\langle v,h^{-1}v\right\rangle \right)\sqrt{\left\langle v,h^{-1}v\right\rangle }+\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{2}\left\langle v,h^{-1}v\right\rangle ^{\frac{3}{2}} \nonumber\\ &\qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{3}\left(\left\langle v,h^{-1}v\right\rangle ^{\frac{5}{2}}+\left\Vert v\right\Vert \left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}+\left\langle v,h^{-\frac{1}{2}}v\right\rangle \left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}\right) \nonumber\\ &\qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{4} \left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}} \left(\sqrt{\left\langle v,h^{-1}v\right\rangle }\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}+\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\right) \nonumber\\ &\qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{5} \left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{\frac{3}{2}} \left(\left\langle v,h^{-1}v\right\rangle ^{\frac{3}{2}}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}+\left\langle v,h^{-1}v\right\rangle \left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\right)\nonumber\\ &\qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{6}\left\langle v,h^{-\frac{1}{2}}v\right\rangle ^{2}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}} \nonumber\\ & \qquad +\left(1+\left\langle v,h^{-1}v\right\rangle \right)^{7}\left\langle v,h^{-\frac{2}{3}}v\right\rangle ^{3}\left\langle v,h^{-\frac{5}{4}}v\right\rangle ^{2}\left\langle v,h^{-\frac{4}{3}}v\right\rangle ^{\frac{3}{2}}. \end{align} $$

The right hand can be simplified further using the Hölder estimates

(9.93) $$ \begin{align} \left\langle v, h^{-\frac 1 2}v\right\rangle \le \|v\| \left\langle v, h^{-1}v\right\rangle^{\frac 1 2},\quad \left\langle v, h^{-\frac 2 3}v\right\rangle^{\frac 3 2} \le \|v\| \left\langle v, h^{-1}v\right\rangle. \end{align} $$

All this gives the following:

Proposition 9.21. For all $t\in \left[0,1\right]$ , it holds that

$$ \begin{align*} \max_{1\leq j\leq n}\left\Vert h^{-\frac{1}{2}}E (t )x_{j}\right\Vert \le C \alpha \left(1+\left\langle v,h^{-1}v\right\rangle \right)^{8} \left( \langle v, h^{-1} v\rangle^{\frac 1 2} + \|v\| \langle v, h^{-\frac 5 4} v\rangle^{2} \right) \left( 1+ \|v\| \langle v, h^{-\frac 4 3} v\rangle^{\frac 3 2} \right), \end{align*} $$

where $\alpha =\max _{1\leq j\leq n}\left\langle v,x_{j}\right\rangle $ and $C>0$ is a constant independent of all quantities.

Conclusion of Proposition 9.6: Inserting $h_k$ and $v_k$ in Proposition 9.7 and (7.11), we have immediately

(9.94) $$ \begin{align} \mathrm{tr} \Big|h_k^{-1/2} (\widetilde{E}_k - h_k) h_k^{-1/2}\Big| \le \langle v_k, h_k^{-1} v_k\rangle \le C \hat V_k. \end{align} $$

Next, consider the corresponding expressions on the right-hand side of Proposition 9.21. Recall that $\alpha _{k}=\max _{p\in L_{k}}\left\langle v_{k},e_{p}\right\rangle \le C(\hat {V}_{k})^{\frac {1}{2}}k_{F}^{-\frac {1}{2}}$ . Moreover, by Propositions A.1, A.2 and A.3, we get

(9.95) $$ \begin{align} \left\langle v_k, h_k^{\beta} v_k\right\rangle &\le C \hat V_k k_F^{-1} \sum_{p\in L_k} \lambda_{k,p}^{\beta} \le C \hat V_k (|k| k_F)^{1+\beta}, \quad 0\ge \beta\ge -\frac 5 4, \nonumber\\ \left\langle v_{k},h_{k}^{\beta}v_{k}\right\rangle^{\frac 3 2} & \le C(\hat{V}_{k})^{\frac 3 2} \left(|k|k_{F}\right)^{-\frac{1}{2}} |k|^{6}\log\left(k_{F}\right),\quad \beta\le -\frac 4 3. \end{align} $$

Putting these bounds together, we deduce from Proposition 9.21 that

(9.96) $$ \begin{align} \max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}E_{k} (t )e_{p}\right\Vert & \leq C(\hat{V}_{k})^{\frac{1}{2}}k_{F}^{-\frac{1}{2}} \left(1+\hat{V}_{k}\right)^8 \left( (\hat V_k)^{\frac 1 2} + (\hat V_k)^{\frac 5 2}\right) \left( 1 + \hat V_k^2 |k|^6 \log k_F\right) \nonumber \\ & \leq Ck_{F}^{-\frac{1}{2}}\left(\hat{V}_{k}+\hat{V}_{k}^{3}|k|^{6}\log\left(k_{F}\right)\right) \end{align} $$

for $|k|\leq k_{F}^{\gamma }$ , as claimed. Similarly, inserting (9.95) in Corollary 9.12 and Proposition 9.14, we see that

(9.97) $$ \begin{align} \left\Vert h^{-\frac{1}{2}}J\right\Vert _{\text{HS}} &\leq C\left(1+\left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle \right)\left\langle v_{k},h_{k}^{-\frac{3}{2}}v_{k}\right\rangle \leq C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\hat{V}_{k}\left(1+\hat{V}_{k}\right)|k|^{3+\frac{2}{3}}, \nonumber\\ \left\Vert h^{-\frac{1}{2}}\left[J,h\right]h^{-\frac{1}{2}}\right\Vert _{\text{HS}} & \leq C\left(1+\left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle \right)^{3}\left(\left\langle v_{k},h_{k}^{-1}v_{k}\right\rangle +\left\langle v_{k},h_{k}^{-\frac{1}{2}}v_{k}\right\rangle \left\langle v_{k},h_{k}^{-\frac{5}{4}}v_{k}\right\rangle ^{2}\right)\nonumber \\ & \leq C\left(1+\hat{V}_{k}\right)^{3}\left(\hat{V}_{k}+\hat{V}_{k}\left(|k|k_{F}\right)^{\frac{1}{2}}\left(\hat{V}_{k}\left(|k|k_{F}\right)^{-\frac{1}{4}}\right)^{2}\right) \leq C\hat{V}_{k}. \end{align} $$

Here, we also note that $\hat V_k$ is uniformly bounded, and hence, the constant C may depend on V, but it is still independent of k and $k_F$ .

9.4 Gronwall estimates for the kinetic operator

We now come to the kinetic Gronwall estimates for the transformation $e^{\mathcal {J}}$ . We have the following:

Proposition 9.22. Assume $\sum _{k\in \mathbb {Z}^{3}}\hat {V}_{k}|k|<\infty $ and $S_C=\mathbb {Z}^3_+ \cap \overline {B}\left(0,k_{F}^{\gamma }\right)$ with $0<\gamma <\frac {1}{47}$ . Then for all $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ and $\left|t\right|\leq 1$ , it holds that

$$ \begin{align*} \left\langle e^{t\mathcal{J}}\Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}e^{t\mathcal{J}}\Psi\right\rangle & \leq C\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \\ \left\langle e^{t\mathcal{J}}\Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}e^{t\mathcal{J}}\Psi\right\rangle & \leq C\left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \end{align*} $$

for a constant $C>0$ independent of $k_{F}$ .

Proof. Write $\Psi _{t}=e^{t\mathcal {J}}\Psi $ for brevity. By the commutator in (9.15), we have

(9.98) $$ \begin{align} -\frac{d}{dt}\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle =\left\langle \Psi_{t},\left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}\right]\Psi_{t}\right\rangle =2\sum_{k\in S_{C}}\left\langle \Psi_{t},\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right)\Psi_{t}\right\rangle \end{align} $$

with $\tilde {Q}_{1}^{k}$ defined in (4.35). Moreover, Proposition 4.8 allows us to estimate

(9.99) $$ \begin{align} \sum_{k\in S_{C}}\left|\left\langle \Psi_{t},\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right)\Psi_{t}\right\rangle \right|\leq\sum_{k\in S_{C}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}}\left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle. \end{align} $$

Since

(9.100) $$ \begin{align} \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}=\left(\begin{array}{cc} h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}} & 0\\ 0 & h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}} \end{array}\right), \end{align} $$

by Proposition 9.6 we can estimate further that

(9.101) $$ \begin{align} \sum_{k\in S_{C}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}} & =\sum_{k\in S_{C}}\left\Vert h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}}\right\Vert _{\text{Op}}\leq\sum_{k\in S_{C}}\left\Vert h_{k}^{-\frac{1}{2}}\left[J_{k},h_{k}\right]h_{k}^{-\frac{1}{2}}\right\Vert _{\text{HS}}\nonumber \\ & \leq C\sum_{k\in S_C}\hat{V}_{k}\leq C. \end{align} $$

Hence, $\left|\frac {d}{dt}\left\langle \Psi _{t},H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi _{t}\right\rangle \right|\leq C\left\langle \Psi _{t},H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi _{t}\right\rangle $ , so by Gronwall’s lemma

(9.102) $$ \begin{align} \left\langle \Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle \leq\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle e^{C\left|t\right|}\leq C\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle ,\quad\left|t\right|\leq1. \end{align} $$

For $\left\langle \Psi _{t},\mathcal {N}_{E}H_{\operatorname {\mathrm {kin}}}^{\prime }\Psi _{t}\right\rangle $ , besides the commutator in (9.15), we also note that

(9.103) $$ \begin{align} \left[\mathcal{J},\mathcal{N}_{E}\right] & =\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left[b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p}),\mathcal{N}_{E}\right]\nonumber \\ & =\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)\left[b_{k}(e_{p}),\mathcal{N}_{E}\right]+\left[b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right),\mathcal{N}_{E}\right]b_{k}(e_{p})\right)\\ & =\sum_{k\in S_{C}}\sum_{p\in L_{k}^{\pm}}\left(b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})-b_{k}^{\ast}\left(J_{k}^{\oplus}e_{p}\right)b_{k}(e_{p})\right)=0.\nonumber \end{align} $$

Here again, we used $\left[\mathcal {N}_{E},b_{k}(\varphi )\right]=-b_{k}$ for all $\varphi \in \ell ^{2}(L_{k}^{\pm })$ , which follows from (1.75) and linearity. Hence,

$$\begin{align*} -\frac{d}{dt}\left\langle \Psi_{t},\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle =\left\langle \Psi_{t},\mathcal{N}_{E}\left[\mathcal{J},H_{\operatorname{\mathrm{kin}}}^{\prime}\right]\Psi_{t}\right\rangle=2\sum_{k\in S_{C}}\left\langle \Psi_{t},\mathcal{N}_{E}\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right)\Psi_{t}\right\rangle. \end{align*}$$

Now, it holds that $[\mathcal {N}_{E},\tilde {Q}_{1}^{k}([J_{k}^{\oplus },h_{k}^{\oplus }])]=0$ (as can be seen by a computation similar to that of equation (9.103)), so we may estimate as above for

(9.104) $$ \begin{align} &\sum_{k\in S_{C}}\left|\left\langle \Psi_{t},\mathcal{N}_{E}\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right)\Psi_{t}\right\rangle \right| =\sum_{k\in S_{C}}\left|\left\langle \mathcal{N}_{E}^{\frac{1}{2}}\Psi_{t},\tilde{Q}_{1}^{k}\left(\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\right)\mathcal{N}_{E}^{\frac{1}{2}}\Psi_{t}\right\rangle \right|\nonumber\\ &\quad \leq\sum_{k\in S_{C}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left[J_{k}^{\oplus},h_{k}^{\oplus}\right]\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}}\left\langle \mathcal{N}_{E}^{\frac{1}{2}}\Psi_{t},H_{\operatorname{\mathrm{kin}}}^{\prime}\mathcal{N}_{E}^{\frac{1}{2}}\Psi_{t}\right\rangle \leq C\left\langle \Psi_{t},\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_{t}\right\rangle , \end{align} $$

where we also used that $[\mathcal {N}_{E},H_{\operatorname {\mathrm {kin}}}^{\prime }]=0$ . The second claim now follows.

10 Conclusion of the main results

Now we are ready to provide the proof of the main theorems stated in the introduction.

10.1 Proof of Theorem 1.1

The proof follows almost immediately by the analysis we have performed throughout the paper, for we will simply take $\mathcal {U}=e^{\mathcal {J}}e^{\mathcal {K}}$ where $e^{\mathcal {K}}$ is the quasi-bosonic Bogolubov transformation $e^{\mathcal {K}}$ of Section 4 and $e^{\mathcal {J}}$ is the second transformation of Section 9.

Step 1: Let us start from the decomposition (1.22):

(10.1) $$ \begin{align} H_N - E_{\mathrm{FS}} = H_{\operatorname{\mathrm{kin}}}' + k_F^{-1} H_{\mathrm{int}}' = H_{\operatorname{\mathrm{kin}}}' + \sum_{k\in S_C} \left(H_{\mathrm{ int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right) + \mathcal{E}_{\mathrm{NB}}, \end{align} $$

where $H_{\text {int}}^{k}$ is given in (1.29), $\mathcal {E}_{\mathrm {NB}}$ is given in (2.22), and $S_{C}=\overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{+}^{3}$ with $0<\gamma < \frac 1 {47}$ . From Proposition 2.4, the non-bosonizable term $\mathcal {E}_{\mathrm {NB}}$ is estimated as

(10.2) $$ \begin{align} \pm \mathcal{E}_{\mathrm{NB}} \le C k_F^{-\gamma/2} (H_{\mathrm{kin}}' + k_F^{-1} \mathcal{N}_E H_{\mathrm{kin}}' + k_F). \end{align} $$

By the Gronwall estimates of Propositions 8.1, 9.22 and the choice $\mathcal {U}=e^{\mathcal {J}}e^{\mathcal {K}}$ , we have

(10.3) $$ \begin{align} \pm \mathcal{U} \mathcal{E}_{\mathrm{NB}} \mathcal{U}^* \le C k_F^{-\gamma/2} (H_{\mathrm{kin}}' + k_F^{-1} \mathcal{N}_E H_{\mathrm{kin}}' + k_F). \end{align} $$

Thus, it remains to apply the transformations $e^{\mathcal K}$ and $e^{\mathcal J}$ to the bosonizable terms.

Step 2: Now we apply the transformation $e^{\mathcal K}$ . By Proposition 5.7, we have

(10.4) $$ \begin{align} & e^{\mathcal{K}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}H_{\operatorname{\mathrm{int}}}^{k}\right)e^{-\mathcal{K}}\nonumber\\ &\quad =H_{\operatorname{\mathrm{kin}}}^{\prime}+\sum_{k\in S_{C}}Q_{1}^{k} (E_k^\oplus - h_k^\oplus) +\sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{K}} \left(\mathcal{E}_{1}^{k} (A_k^{\oplus}(t)) +\mathcal{E}_{2}^{k} (B_k^\oplus(t))\right)e^{-(1-t)\mathcal{K}}\,dt. \end{align} $$

We will use the kinetic estimate of Proposition 6.5 and the Gronwall estimates of Proposition 8.1 to bound the exchange terms in (10.4). Thanks to the one-body estimates in Propositions 7.3, 7.2 and our assumption $\sum _{k\in S_{C}}\hat {V}_{k}|k|<\infty $ , we get

(10.5) $$ \begin{align} \sum_{k\in S_{C}}\max_{t\in\left[0,1\right]}\left\{ \max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}A_{k}^{\oplus} (t )e_{p}\right\Vert ,\max_{p\in L_{k}}\left\Vert h_{k}^{-\frac{1}{2}}B_{k}^{\oplus} (t )e_{p}\right\Vert \right\} &\leq C\sum_{k\in S_{C}}k_{F}^{-\frac{1}{2}}\hat{V}_{k}\left(1+\hat{V}_{k}^{2}\right)\leq Ck_{F}^{-\frac{1}{2}}, \nonumber\\ \sum_{k\in S_{C}}\max_{t\in\left[0,1\right]}\left\{ \left\Vert A_{k}^{\oplus} (t )\right\Vert _{\infty,2},\left\Vert B_{k}^{\oplus} (t )\right\Vert _{\infty,2}\right\} \leq C\sum_{k\in S_{C}}&\hat{V}_{k}|k|^{\frac{1}{2}}\left(1+\hat{V}_{k}\right)\leq C, \nonumber\\ \sum_{k\in S_{C}}\left(\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}K_{k}^{\oplus}\right\Vert _{\text{HS}}+\left\Vert K_{k}^{\oplus}\right\Vert _{\infty,2}\right) \leq C(\log k_{F})^{\frac{2}{3}}& k_{F}^{-\frac{1}{3}}\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}. \end{align} $$

All this gives that for every state $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ and $\Psi _t=e^{-(1-t)\mathcal {K}}\Psi $ ,

(10.6) $$ \begin{align} & \sum_{k\in S_{C}}\int_{0}^{1}\left|\left\langle \Psi_t,\left(\mathcal{E}_{1}^{k} (A^{\oplus}_k(t) )+\mathcal{E}_{2}^{k} (B^{\oplus}_k(t) )\right) \Psi_t \right\rangle \right|dt \leq C(\log k_{F})^{\frac{2}{3}}\left(\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}\right)\nonumber\\ &\quad \times\left(k_{F}^{-\frac{5}{6}}\max_{t\in\left[0,1\right]}\sqrt{\left\langle \Psi_t,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t\right\rangle \left\langle \Psi_t,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t\right\rangle }\right.\nonumber\\ &\quad\left. +\ k_{F}^{-\frac{1}{3}}\max_{t\in\left[0,1\right]}\left\langle \Psi_t,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t\right\rangle +k_{F}^{-\frac{1}{3}}\max_{t\in\left[0,1\right]}\sqrt{\left\langle \Psi_t,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t\right\rangle }\right)\nonumber \\ & \leq C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\left(\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}\right)\left(\left\langle \Psi,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle +k_{F}^{-1}\left\langle \Psi,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle +k_{F}\left\Vert \Psi\right\Vert ^{2}\right), \end{align} $$

where we also used the Cauchy–Schwarz inequality to split the square roots at the end. Thus, the exchange terms in (10.4) can be estimated as

(10.7) $$ \begin{align} &\pm \sum_{k\in S_{C}}\int_{0}^{1} e^{(1-t)\mathcal{K}}\left(\mathcal{E}_{1}^{k} (A^{\oplus}_k(t) )+\mathcal{E}_{2}^{k} (B^{\oplus}_k(t) )\right) e^{-(1-t)\mathcal{K}} dt \nonumber\\ & \leq C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\left(\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}\right)\left( H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1} \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} +k_{F}\right). \end{align} $$

It remains to consider the main term $Q_1(E_k^{\oplus } - h_k^{\oplus })$ on the right side of (10.4). We use the normal order form in (4.34):

(10.8) $$ \begin{align} \sum_{k\in S_C} Q_1( E_k^{\oplus} - h_k^{\oplus})= \sum_{k\in S_C} 2\widetilde{Q}_1 ( E_k^{\oplus} - h_k^{\oplus}) + \sum_{k\in S_C} 2\mathrm{tr} ( E_k -h_k)+ \sum_{k\in S_C} \varepsilon_k(E_k^{\oplus} -h_k^\oplus). \end{align} $$

By Propositions 4.9, 7.2 and 2.1,

(10.9) $$ \begin{align} \pm\sum_{k\in S_{C}}\varepsilon_{k}\left( E_{k}^{\oplus}-h_{k}^{\oplus}\right)\leq C \sum_{k\in S_C} k_{F}^{-1}\hat{V}_{k}\left(1+\hat{V}_{k}\right) \mathcal{N}_{E} \le Ck_{F}^{-1} H_{\operatorname{\mathrm{kin}}}'. \end{align} $$

Moreover, by Proposition 7.1, we have

(10.10) $$ \begin{align} \sum_{k\in S_C} \left( 2\operatorname{\mathrm{tr}}\left({E}_{k}-h_{k}\right)-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right| \right) & = \sum_{k\in S_C} \frac{2}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt \end{align} $$

with $F (x )=\log \left(1+x\right)-x$ . Thus in summary, we conclude from (10.4) that

(10.11) $$ \begin{align} & e^{\mathcal{K}} \left( H_{\operatorname{\mathrm{kin}}}' + \sum_{k\in S_C} \left(H_{\mathrm{ int}}^{k}-\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\left|L_{k}\right|\right) \right) e^{-\mathcal{K}} \nonumber\\ &= H_{\operatorname{\mathrm{kin}}}^{\prime}+ 2\sum_{k\in S_{C}} \widetilde Q_{1}^{k} (E_{k}^{\oplus}-h_{k}^{\oplus}) + \sum_{k\in S_C} \frac{2}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt + \mathcal{E}_{\mathcal K}, \end{align} $$

where

(10.12) $$ \begin{align} \pm \mathcal{E}_{\mathcal K} \le C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\left(\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}\right)\left( H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1} \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} +k_{F}\right). \end{align} $$

Step 3: Next, we apply the transformation $e^{\mathcal J}$ to the right-hand side of (10.11). From (10.12) and the Gronwall estimates of Proposition 9.22, we have

(10.13) $$ \begin{align} \pm e^{\mathcal{J}} \mathcal{E}_{\mathcal K} e^{-\mathcal{J}} \le C(\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}\left(\sum_{k\in S_C}\hat{V}_{k}|k|^{3+\frac{2}{3}}\right)\left( H_{\operatorname{\mathrm{kin}}}^{\prime}+k_{F}^{-1} \mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} +k_{F}\right). \end{align} $$

For the main terms, by Proposition 9.1,

(10.14) $$ \begin{align} &e^{\mathcal{J}}\left(H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left(E_{k}^{\oplus}-h_{k}^{\oplus}\right)\right)e^{-\mathcal{J}} \\ & =H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( \widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) +2\sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{J}} {\mathcal{E}}_{3}^{k}(F_{k}^{\oplus}(t)) e^{-(1-t)\mathcal{J}}dt .\nonumber \end{align} $$

Let us bound the exchange term $\mathcal {E}_3(\cdot )$ . For all $k\in \overline {B}\left(0,k_{F}^{\gamma }\right)\cap \mathbb {Z}_{\ast }^{3}$ with $0<\gamma <\frac {1}{47}$ and $t\in [0,1]$ , by Proposition 9.6, we have

(10.15) $$ \begin{align} \max_{p\in L_{k}^{\pm}}\left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}E_k(t) e_{p}\right\Vert &\le Ck_{F}^{-\frac{1}{2}}\left(\hat{V}_{k}+\hat{V}_{k}^{3}|k|^{6}\log\left(k_{F}\right)\right) \le Ck_{F}^{-\frac{1}{2}}\left(\hat{V}_{k}+\hat{V}_{k}^{3}|k|^{3} k_F^{\frac 3 {47}}\log\left(k_{F}\right)\right) ,\nonumber\\ \sum_{l\in S_{C}}\left\Vert \left(h_{l}^{\oplus}\right)^{-\frac{1}{2}}J_{l}^{\oplus}\right\Vert _{\operatorname{\mathrm{HS}}} &\le C (\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}} \sum_{l\in S_{C}} \hat{V}_{l}\left(1+\hat{V}_{l}\right) \le C (\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}}. \end{align} $$

Hence, using the kinetic estimate of Proposition 9.5, Gronwall’s bounds of Proposition 9.22 and the assumption $\sum _{k\in \mathbb {Z}^3_{*}} \hat V_k |k| <\infty $ , we find that for every state $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ and $\Psi _t=e^{-(1-t)\mathcal {J}}\Psi $ ,

(10.16) $$ \begin{align} &\sum_{k\in S_{C}}\int_{0}^{1} \left| \langle \Psi_t, {\mathcal{E}}_{3}^{k}(F_{k}^{\oplus}(t)) \Psi_t\rangle \right| dt \\ &\le \sum_{k\in S_{C}} C (\log k_{F})^{\frac{2}{3}}k_{F}^{-\frac{1}{3}} k_{F}^{-\frac{1}{2}}\left(\hat{V}_{k}+\hat{V}_{k}^{3}|k|^{3}k_F^{\frac 3 {47}}\log\left(k_{F}\right)\right) \max_{t\in [0,1]} \sqrt{\left\langle \Psi_t,H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t\right\rangle \left\langle \Psi_t,\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi_t \right\rangle} \nonumber\\ &\le C(\log k_F)^{\frac 5 3} k_F^{-\frac{1}{3}} \left\langle \Psi, ( k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime}+k_F ) \Psi \right\rangle. \nonumber \end{align} $$

Here, we used $\sum _{k\in \mathbb {Z}^3_{*}} \hat V_k^3 |k|^3 \le \left( \sum _{k\in \mathbb {Z}^3_{*}} \hat V_k |k| \right)^3 <\infty $ . Consequently,

(10.17) $$ \begin{align} \pm \sum_{k\in S_{C}}\int_{0}^{1}e^{(1-t)\mathcal{J}} {\mathcal{E}}_{3}^{k}(F_{k}^{\oplus}(t)) e^{-(1-t)\mathcal{J}}dt \le C (\log k_F)^{\frac 5 3} k_F^{-\frac{1}{3}} (k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime}+k_F ). \end{align} $$

In summary, we have for $\mathcal {U}= e^{\mathcal {J}} e^{\mathcal {K}}$ and $0<\gamma <\frac 1 {47}$ ,

(10.18) $$ \begin{align} \mathcal{U} H_N \mathcal{U}^* = E_{\mathrm{FS}} + H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in S_{C}}\tilde{Q}_{1}^{k}\left( \widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) + \sum_{k\in S_C} \frac{2}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt + \mathcal{E}_{\mathcal J}, \end{align} $$

where the error term is collected from (10.3), (10.13), (10.17) which satisfies

(10.19) $$ \begin{align} \pm \mathcal{E}_{\mathcal J} \le C k_F^{-\gamma/2} (k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime}+k_F ). \end{align} $$

Step 4: Finally, let us remove the cutoff $S_C= \mathbb {Z}_{+}^{3}\cap \overline {B}\left(0,k_{F}^{\gamma }\right) $ on the right-hand side of (10.18). By Proposition 7.1, we can bound

(10.20) $$ \begin{align} \left|\frac{1}{\pi}\sum_{k\in\mathbb{Z}_{\ast}^{3}\backslash S_C}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt\right|\leq Ck_{F}\sum_{k\in\mathbb{Z}_{\ast}^{3}\backslash S_C}\hat{V}_{k}^{2}|k| \le C k_F^{1 - \gamma}. \end{align} $$

Here, we used $\sum _{k\in \mathbb {Z}^3_{*}} \hat V_k^2 |k|^2 \le \left( \sum _{k\in \mathbb {Z}^3_{*}} \hat V_k |k| \right)^2 <\infty $ . Moreover, by Propositions 4.8 and 9.6 (together with the fact that the trace norm dominates the operator norm), we can bound

(10.21) $$ \begin{align} \pm\tilde{Q}_{1}^{k}\left(\widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) &\leq \left\Vert \left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\left(\widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right)\left(h_{k}^{\oplus}\right)^{-\frac{1}{2}}\right\Vert _{\text{Op}}H_{\operatorname{\mathrm{kin}}}^{\prime} \nonumber\\ &= \left\Vert h_{k}^{-\frac{1}{2}}\left(\widetilde{E}_{k} -h_{k} \right) h_{k}^{-\frac{1}{2}}\right\Vert _{\text{Op}}H_{\operatorname{\mathrm{kin}}}^{\prime} \le C \hat V_k H_{\operatorname{\mathrm{kin}}}^{\prime} \end{align} $$

for all $k\in \mathbb {Z}^3_+$ , and hence,

(10.22) $$ \begin{align} \pm \sum_{k\in \mathbb{Z}^3_+\backslash S_C} \tilde{Q}_{1}^{k}\left(\widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) \le C \left( \sum_{k\in \mathbb{Z}^3_+\backslash S_C} \hat V_k \right) H_{\operatorname{\mathrm{kin}}}^{\prime} \le C k_F^{-\gamma}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

Therefore, we can deduce from (10.18) that for $\mathcal {U}= e^{\mathcal {J}} e^{\mathcal {K}}$ and $0<\gamma <\frac 1 {47}$ ,

(10.23) $$ \begin{align} \mathcal{U} H_N \mathcal{U}^* =E_{\mathrm{FS}} + H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in \mathbb{Z}^3_+}\tilde{Q}_{1}^{k}\left( \widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) + \sum_{k\in \mathbb{Z}^3_{*}} \frac{1}{\pi}\int_{0}^{\infty}F\left(\frac{\hat{V}_{k}k_{F}^{-1}}{(2\pi)^{3}}\sum_{p\in L_{k}}\frac{\lambda_{k,p}}{\lambda_{k,p}^{2}+t^{2}}\right)dt + \mathcal{E}_{\mathcal U}, \end{align} $$

where

(10.24) $$ \begin{align} \pm \mathcal{E}_{\mathcal U} \le C k_F^{-\gamma/2} (k_F^{-1}\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime}+k_F ). \end{align} $$

The statement of Theorem 1.1 follows by recognizing the identity

$$ \begin{align*}2\sum_{k\in \mathbb{Z}^3_+}\tilde{Q}_{1}^{k}\left( \widetilde{E}_{k}^{\oplus}-h_{k}^{\oplus}\right) = 2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\left\langle e_{p},\left(\widetilde{E}_{k}-h_{k}\right)e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}, \end{align*} $$

which follows from the definition of $\tilde {Q}_{1}^{k}$ in (4.35).

10.2 Proof of Theorem 1.2

Let $\Psi \in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ be a normalized eigenstate of $H_{N}$ with energy $\left\langle \Psi ,H_{N}\Psi \right\rangle \le E_{\mathrm {FS}} + \kappa k_F$ for some $\kappa>0$ . Denoting $\tilde {H}_{N}=H_{N}- E_{\mathrm {FS}}$ , we have $\tilde {H}_{N}\Psi =E'\Psi $ with $E' \le \kappa k_F$ . Using (1.22) and the obvious inequality $A^*A\ge 0$ , we obtain the Onsager-type estimate

(10.25) $$ \begin{align} \tilde{H}_{N} - H_{\operatorname{\mathrm{kin}}}^{\prime} &= \frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left(\text{d}\Gamma(e^{-ik\cdot x})^{\ast}\text{d}\Gamma(e^{-ik\cdot x})-\left|L_{k}\right|\right)\\ & \geq -\frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}_{\ast}^{3}}\hat{V}_{k}\left|L_{k}\right| \ge - Ck_{F} \sum_{k\in \mathbb{Z}^3} |k| \hat V_k.\nonumber \end{align} $$

Here, we used $\left|L_{k}\right|\leq Ck_{F}^{2}|k|$ for all $k\in \mathbb {Z}_{\ast }^{3}$ (see Proposition A.1). From (10.25) and the assumption $\tilde {H}_{N}\Psi =E'\Psi $ with $E' \le \kappa k_F$ , we deduce immediately that

(10.26) $$ \begin{align} \langle \Psi, H_{\mathrm{kin}}' \Psi\rangle \le C (\kappa+1) k_F. \end{align} $$

To prove the bound for $\mathcal {N}_E H_{\mathrm {kin}}'$ , we use the operator inequality

(10.27) $$ \begin{align} \mathcal{N}_{E}^{2}H_{\operatorname{\mathrm{kin}}}^{\prime}&=\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\mathcal{N}_{E}\leq\mathcal{N}_{E}\tilde{H}_{N}\mathcal{N}_{E}+Ck_{F}\mathcal{N}_{E}^{2} \nonumber\\ &= \frac{1}{2}\left(\mathcal{N}_{E}^{2}\tilde{H}_{N}+\tilde{H}_{N}\mathcal{N}_{E}^{2}-\left[\mathcal{N}_{E},\left[\mathcal{N}_{E},\tilde{H}_{N}\right]\right]\right)+Ck_{F}\mathcal{N}_{E}^{2}, \end{align} $$

which follows from (10.25) and the fact that $\left[\mathcal {N}_{E},H_{\operatorname {\mathrm {kin}}}^{\prime }\right]=0$ . Thanks to the eigenvalue equation $\tilde {H}_{N}\Psi =E'\Psi $ with $E'\le \kappa k_F$ , we deduce that

(10.28) $$ \begin{align} \left\langle \Psi,\mathcal{N}_{E}^{2}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \leq C(\kappa+1) k_{F} \left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle -\frac{1}{2}\left\langle \Psi,\left[\mathcal{N}_{E},\left[\mathcal{N}_{E},\tilde{H}_{N}\right]\right]\Psi\right\rangle. \end{align} $$

Using $\mathcal {N}_{E} = \sum _{s\in B_F^c} c_{s}^* c_s$ and

(10.29) $$ \begin{align} [c_{s}^* c_s, c_{p+k}^{\ast}c_{q-k}^{\ast}c_{q}c_{p}] = c_{p+k}^{\ast}c_{q-k}^{\ast}c_{q}c_{p} (\delta_{s,p+k} + \delta_{s,q-k}- \delta_{s,q}- \delta_{s,p}), \end{align} $$

we deduce from (1.8) that

(10.30) $$ \begin{align} \left[\mathcal{N}_{E},\left[\mathcal{N}_{E},\tilde{H}_{N}\right]\right] =\frac{k_{F}^{-1}}{2(2\pi)^{3}}\sum_{k\in\mathbb{Z}^{3}} \sum_{p,q\in\mathbb{Z}^{3}} \hat{V}_{k} c_{p+k}^{\ast}c_{q-k}^{\ast}c_{q}c_{p} \Big( \sum_{s\in B_F^c} (\delta_{s,p+k} + \delta_{s,q-k}- \delta_{s,q}- \delta_{s,p}) \Big)^2. \end{align} $$

Using the obvious bound

(10.31) $$ \begin{align} 0\le \Big( \sum_{s\in B_F^c} (\delta_{s,p+k} + \delta_{s,q-k}- \delta_{s,q}- \delta_{s,p}) \Big)^2 \le 4 \end{align} $$

and the Cauchy–Schwarz inequality, we estimate

(10.32) $$ \begin{align} &\left| \left\langle \Psi,\left[\mathcal{N}_{E},\left[\mathcal{N}_{E},\tilde{H}_{N}\right]\right]\Psi\right\rangle \right| \le C k_F^{-1} \sum_{k\in\mathbb{Z}^{3}} \sum_{p,q\in\mathbb{Z}^{3}} \hat V_k \| c_{p+k} c_{q-k} \Psi\| \|c_{q}c_{p} \Psi\| \nonumber \\ &\le C k_F^{-1} \sum_{k\in\mathbb{Z}^{3}} \hat V_k \sum_{p,q\in\mathbb{Z}^{3}} ( \| c_{p+k} c_{q-k} \Psi\|^2 + \|c_{q}c_{p} \Psi\|^2) \le C k_F^{-1} \sum_{k\in\mathbb{Z}^{3}} \hat V_k \langle \Psi, \mathcal{N}_E^2 \Psi\rangle. \end{align} $$

Since $\hat V$ is summable, (10.28) and (10.32) imply that

(10.33) $$ \begin{align} \left\langle \Psi,\mathcal{N}_{E}^{2}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle \leq C(\kappa+1) k_{F} \left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle. \end{align} $$

Combining with the inequality $H_{\operatorname {\mathrm {kin}}}^{\prime } \ge \mathcal {N}_{E}$ from Proposition 2.1, we deduce by Hölder’s inequality

(10.34) $$ \begin{align} \left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle \le \left\langle \Psi,\mathcal{N}_{E}^{3} \Psi\right\rangle^{2/3} \le \left\langle \Psi,\mathcal{N}_{E}^{2}H_{\operatorname{\mathrm{kin}}}^{\prime}\Psi\right\rangle^{2/3} \leq \Big( C(\kappa+1) k_{F} \left\langle \Psi,\mathcal{N}_{E}^{2}\Psi\right\rangle \Big)^{2/3}, \end{align} $$

which implies that $\left\langle \Psi ,\mathcal {N}_{E}^{2} \Psi \right\rangle \le C (\kappa +1)^2k_F^2$ , and hence by (10.33) again,

(10.35) $$ \begin{align} \quad \langle \Psi, \mathcal{N}_E^2 H_{\mathrm{kin}}' \Psi\rangle \le C (\kappa+1)^3 k_F^3. \end{align} $$

The bound $\left\langle \Psi ,\mathcal {N}_{E} H_{\operatorname {\mathrm {kin}}}' \Psi \right\rangle \le C(\kappa +1)^2k_F^2$ follows from (10.26) and (10.35). In summary, we have

(10.36) $$ \begin{align} \left\langle \Psi, (k_F^{-1}\mathcal{N}_{E} H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \Psi\right\rangle \leq C (\kappa +1 )^2 k_F. \end{align} $$

By the Gronwall estimates of Propositions 8.1, 9.22 and the choice $\mathcal {U}=e^{\mathcal {J}}e^{\mathcal {K}}$ , we also obtain

(10.37) $$ \begin{align} \left\langle \mathcal{U} \Psi, (k_F^{-1}\mathcal{N}_{E} H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \mathcal{U} \Psi\right\rangle \leq C (\kappa +1 )^2 k_F. \end{align} $$

10.3 Proof of Theorem 1.2

Taking the expectation against $\Psi _{\mathrm {FS}}$ of the operator estimate in Theorem 1.1, we have

(10.38) $$ \begin{align} \inf \sigma(H_N) =\inf \sigma(\mathcal{U} H_N\mathcal{U}^*) \le \left\langle \Psi_{\mathrm{FS}},\mathcal{U} H_N\mathcal{U}^* \Psi_{\mathrm{FS}}\right\rangle = E_{\mathrm{FS}} + E_{\mathrm{corr}} + O(k_F^{1-\frac{1}{94}+\epsilon}). \end{align} $$

Here, we used the bound on $\mathcal {E}_U$ from Theorem 1.1 and the identities $H_{\operatorname {\mathrm {kin}}}' \Psi _{\mathrm {FS}} = H_{\mathrm {eff}} \Psi _{\mathrm {FS}} =0$ .

To see the lower bound, let $\Psi _{\mathrm {GS}}\in D\left(H_{\operatorname {\mathrm {kin}}}^{\prime }\right)$ be the normalized ground state of $H_{N}$ . By the definition of $\Psi _{\mathrm {GS}}$ and the above upper bound, we have

(10.39) $$ \begin{align} \left\langle \Psi_{\mathrm{GS}},H_{N} \Psi_{\mathrm{GS}} \right\rangle = \inf \sigma(H_N)\le E_{\mathrm{FS}} + Ck_F, \end{align} $$

and hence, Theorem 1.2 implies that the state $\Psi _{\mathrm {GS}}'= \mathcal {U} \Psi _{\mathrm { GS}}$ satisfies

(10.40) $$ \begin{align} \left\langle \Psi_{\mathrm{GS}}', (k_F^{-1}\mathcal{N}_{E} H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \Psi_{\mathrm{GS}}' \right\rangle \leq C k_F. \end{align} $$

Taking the expectation against $\Psi _{\mathrm {GS}}'$ of the operator estimate in Theorem 1.1, we conclude that

(10.41) $$ \begin{align} \inf \sigma(H_N) &= \left\langle \Psi_{\mathrm{GS}},H_{N} \Psi_{\mathrm{GS}} \right\rangle = \left\langle \Psi_{\mathrm{GS}}',\mathcal{U} H_{N} \mathcal{U} ^* \Psi_{\mathrm{GS}}' \right\rangle\\ &=E_{\mathrm{FS}} + E_{\mathrm{corr}} + \left\langle \Psi_{\mathrm{GS}}', \left( H_{\operatorname{\mathrm{kin}}}^{\prime}+ H_{\mathrm{eff}} +\mathcal{E}_{\mathcal U} \right) \Psi_{\mathrm{GS}}' \right\rangle \ge E_{\mathrm{FS}} + E_{\mathrm{corr}} + O(k_F^{1-\frac{1}{94}+\epsilon}). \nonumber \end{align} $$

Here, we used the operator inequalities

(10.42) $$ \begin{align} H_{\operatorname{\mathrm{kin}}}^{\prime}\ge 0,\quad H_{\mathrm{eff}}\ge 0, \quad \mathcal{E}_{\mathcal U} \ge C k_F^{1-\frac{1}{94}+\epsilon} (k_F^{-1}\mathcal{N}_{E} H_{\operatorname{\mathrm{kin}}}^{\prime} + H_{\operatorname{\mathrm{kin}}}^{\prime} + k_F) \end{align} $$

and the a priori estimate (10.40). This completes the proof of Theorem 1.3.

10.4 Proof of Theorems 1.4 and 1.5

In this subsection, we study the effective operator $H_{\mathrm {eff}}$ in Theorem 1.1 in more detail. First, we prove the following remarkable fact.

Proposition 10.1. We have the operator identity on $D(H_{\operatorname {\mathrm {kin}}}^{\prime })$ :

$$\begin{align*}2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}=\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align*}$$

Proof of Proposition 10.1 .

The idea is simply to interchange the summation on $k\in \mathbb {Z}_{\ast }^{3}$ and $p\in L_{k}$ . By rephrasing the condition that $p\in L_{k}$ , we have the equivalences

(10.43) $$ \begin{align} \left(k\in\mathbb{Z}_{\ast}^{3}\right)\wedge\left(p\in L_{k}\right) & \Leftrightarrow\left(k\in\mathbb{Z}_{\ast}^{3}\right)\wedge\left(|p-k|\leq k_{F}<|p|\right)\nonumber \\ & \Leftrightarrow\left(k\in\mathbb{Z}_{\ast}^{3}\right)\wedge\left(k\in\overline{B}\left(p,k_{F}\right)\right)\wedge\left(p\in B_{F}^{c}\right)\\ & \Leftrightarrow\left(p\in B_{F}^{c}\right)\wedge\left(k\in\overline{B}\left(p,k_{F}\right)\cap\mathbb{Z}^{3}\right),\nonumber \end{align} $$

where we could replace $\mathbb {Z}_{\ast }^{3}$ by $\mathbb {Z}^{3}$ in the last line as the conditions $p\in B_{F}^{c}=\mathbb {Z}^{3}\backslash \overline {B}(0,k_{F})$ and $k\in \overline {B}\left(p,k_{F}\right)$ exclude $k=0$ automatically. Recognizing that $\overline {B}\left(p,k_{F}\right)\cap \mathbb {Z}^{3}=B_{F}+p$ , we can now write

(10.44) $$ \begin{align} 2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p} & =\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\left(|p|^{2}-|p-k|^{2}\right)b_{k,p}^{\ast}b_{k,p}\\ & =\sum_{p\in B_{F}^{c}}\sum_{k\in\left(B_{F}+p\right)}|p|^{2}b_{k,p}^{\ast}b_{k,p}-\sum_{p\in B_{F}^{c}}\sum_{k\in\left(B_{F}+p\right)}|p-k|^{2}b_{k,p}^{\ast}b_{k,p},\nonumber \end{align} $$

and by expanding the excitation operators, we find for the first sum that

(10.45) $$ \begin{align} \sum_{p\in B_{F}^{c}}\sum_{k\in\left(B_{F}+p\right)}|p|^{2}b_{k,p}^{\ast}b_{k,p} & =\sum_{p\in B_{F}^{c}}\sum_{k\in\left(B_{F}+p\right)}|p|^{2}c_{p}^{\ast}c_{p-k}c_{p-k}^{\ast}c_{p} \nonumber \\ &=\sum_{p\in B_{F}^{c}}\left(\sum_{k\in\left(B_{F}+p\right)}c_{p-k}c_{p-k}^{\ast}\right)|p|^{2}c_{p}^{\ast}c_{p}\nonumber \\ & =\sum_{p\in B_{F}^{c}}\left(\sum_{k\in B_{F}}c_{-k}c_{-k}^{\ast}\right)|p|^{2}c_{p}^{\ast}c_{p}=\mathcal{N}_{E}\sum_{p\in B_{F}^{c}}|p|^{2}c_{p}^{\ast}c_{p} \end{align} $$

as $\sum _{k\in B_{F}}c_{-k}c_{-k}^{\ast }=\sum _{k\in B_{F}}c_{k}c_{k}^{\ast }=\mathcal {N}_{E}$ by the particle-hole symmetry, and similarly,

(10.46) $$ \begin{align} \sum_{p\in B_{F}^{c}}\sum_{k\in\left(B_{F}+p\right)}|p-k|^{2}b_{k,p}^{\ast}b_{k,p} & =\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}\sum_{k\in\left(B_{F}+p\right)}|p-k|^{2}c_{p-k}c_{p-k}^{\ast}\\ & =\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}\sum_{k\in B_{F}}|k|^{2}c_{k}c_{k}^{\ast}=\mathcal{N}_{E}\sum_{k\in B_{F}}|k|^{2}c_{k}c_{k}^{\ast}\nonumber \end{align} $$

for the claimed equality of

(10.47) $$ \begin{align} T= 2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p}=\mathcal{N}_{E}\left(\sum_{p\in B_{F}^{c}}|p|^{2}c_{p}^{\ast}c_{p}-\sum_{p\in B_{F}}|p|^{2}c_{p}c_{p}^{\ast}\right)=\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

To complete the proof, let us show that the relevant operators are well defined on the domain $D(H_{\mathrm {kin}}')$ . This is clear for $\mathcal {N}_{E}H_{\operatorname {\mathrm {kin}}}^{\prime }$ since $\mathcal {N}_{E}$ is a bounded operator ( $0\le \mathcal {N}_E\le N$ on $\mathcal {H}_N$ ). For T, we can interchange the summations of k and p using the same observation in (10.43). This gives the quadratic form estimate

(10.48) $$ \begin{align} T&=2\sum_{k\in \mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\lambda_{k,p}b_{k,p}^{\ast}b_{k,p} =\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\vert|p|^{2}-\zeta\vert\,b_{k,p}^{\ast}b_{k,p}+\sum_{k\in \mathbb{Z}_{\ast}^{3}}\sum_{p\in L_{k}}\vert|p-k|^{2}-\zeta\vert\,b_{k,p}^{\ast}b_{k,p} \nonumber\\ &\le \sum_{p\in B_{F}^{c}}\left(\sum_{k\in B_{F}}c_{k}c_{k}^{\ast}\right)\vert|p|^{2}-\zeta\vert\,c_{p}^{\ast}c_{p}+\sum_{p\in B_{F}^{c}}c_{p}^{\ast}c_{p}\sum_{k\in B_{F}}\vert|k|^{2}-\zeta\vert\,c_{k}c_{k}^{\ast}\le \mathcal{N}_E H_{\mathrm{kin}}', \end{align} $$

where $\zeta>0$ is the constant in (1.14). Moreover, it is easily seen that T commutes with both $\mathcal {N}_E$ and $H_{\operatorname {\mathrm {kin}}}^{\prime }$ . Therefore, the above quadratic form estimate also implies the stronger estimate

(10.49) $$ \begin{align} T^{2}\leq\left(\mathcal{N}_{E}H_{\operatorname{\mathrm{kin}}}^{\prime}\right)^{2}, \end{align} $$

which justifies that $D(T)\subset D(\mathcal {N}_{E}H^{\prime }_{\operatorname {\mathrm {kin}}}) \subset D(H_{\operatorname {\mathrm {kin}}}^{\prime })$ .

Now we are ready to give the

Proof of Theorem 1.5.

Thanks to Proposition 10.1 and the identity $\langle e_p, h_k e_q\rangle =\lambda _{k,p}\delta _{p,q}$ , we have

(10.50) $$ \begin{align} H_{\text{eff}}&=H_{\operatorname{\mathrm{kin}}}^{\prime}+2\sum_{k\in\mathbb{Z}^{3}_{*} }\sum_{p,q\in L_{k}}\left\langle e_{p},\left(\widetilde{E}_{k}-h_{k}\right)e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q} \nonumber\\ &=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\langle e_{p},\widetilde{E}_{k}e_{q}\rangle b_{k,p}^{\ast}b_{k,q}-\left(\mathcal{N}_{E}-1\right)H_{\operatorname{\mathrm{kin}}}^{\prime}. \end{align} $$

Since $\left[H_{\text {eff}},\mathcal {N}_{E}\right]=0$ , we can restrict $H_{\text {eff}}$ to the eigenspaces of $\mathcal {N}_{E}$ : for every $M=\{1,2,...\}$ , we can write the restriction to $\left\{ \mathcal {N}_{E}=M\right\} $ for $M\in \mathbb {N}$ in the quasi-bosonic form

(10.51) $$ \begin{align} \left.H_{\text{eff}}\right|_{\mathcal{N}_{E}=M}=2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\left\langle e_{p},\left(\widetilde{E}_{k}-\left(1-M^{-1}\right)h_{k}\right)e_{q}\right\rangle b_{k,p}^{\ast}b_{k,q}. \end{align} $$

Proof of Theorem 1.4.

We only need to verify the statement on the effective operator $\left.H_{\text {eff}}\right|_{\mathcal {N}_{E}=M}$ with $M=1$ . In this case, it is convenient to introduce the total momentum $P=\left(P_{1},P_{2},P_{3}\right)$ , where each $P_{j}$ is given by $P_{j}=\sum _{p\in \mathbb {Z}^{3}}p_{j}c_{p}^{\ast }c_{p}.$ It is easily checked that $P_{j}$ obeys the commutators

(10.52) $$ \begin{align} \left[P_{j},b_{k,p}\right]=-k_{j}b_{k,p},\quad\left[P_{j},b_{k,p}^{\ast}\right]=k_{j}b_{k,p}^{\ast}, \end{align} $$

and additionally $\left[P_{j},H_{\operatorname {\mathrm {kin}}}^{\prime }\right]=0$ , whence the effective Hamiltonian $H_{\text {eff}}$ also commutes with $P_{j}$ , $j=1,2,3$ . It also holds that $\left[\mathcal {N}_{E},P_{j}\right]=0$ , so we may restrict $H_{\text {eff}}$ to the simultanous eigenspaces of $\mathcal {N}_{E}$ and P. It follows from $\left[P_{j},b_{k,p}^{\ast }\right]=k_{j}b_{k,p}^{\ast }$ that this simultaneous eigenspace is precisely

(10.53) $$ \begin{align} \left\{ \Psi\in\mathcal{H}_{N}\mid\mathcal{N}_{E}\Psi=\Psi,\,P\Psi=k\Psi\right\} =\operatorname{\mathrm{span}}\left(b_{k,p}^{\ast}\psi_{\mathrm{FS}}\right)_{p\in L_{k}}=\left\{ b_{k}^{\ast}(\varphi )\psi_{\mathrm{FS}}\mid\varphi\in L^{2}(L_{k})\right\}. \end{align} $$

In fact, the mapping $U:\varphi \mapsto b_{k}^{\ast }(\varphi )\psi _{\mathrm {FS}}$ is an isomorphism. To see that, we compute, using the commutation relations of the excitation operators and the fact that $b_{k}(\phi )\psi _{\mathrm { FS}}=0=\varepsilon _{k,k}(\phi ;\varphi )\psi _{\mathrm {FS}}$ for any $\phi ,\varphi \in L^{2}(L_{k})$ , that

(10.54) $$ \begin{align} \left\langle U\phi,U\varphi\right\rangle & =\left\langle b_{k}^{\ast}(\phi )\psi_{\mathrm{ FS}},b_{k}^{\ast}(\varphi )\psi_{\mathrm{FS}}\right\rangle =\left\langle \psi_{\mathrm{ FS}},\left(b_{k}^{\ast}(\varphi )b_{k}(\phi )+\left\langle \phi,\varphi\right\rangle +\varepsilon_{k,k}\left(\phi;\varphi\right)\right)\psi_{\mathrm{FS}}\right\rangle \\ & =\left\langle \phi,\varphi\right\rangle \left\langle \psi_{\mathrm{FS}},\psi_{\mathrm{FS}}\right\rangle =\left\langle \phi,\varphi\right\rangle ,\nonumber \end{align} $$

so U is a unitary embedding of $L^{2}(L_{k})$ into $\left\{ \Psi \in \mathcal {H}_{N}\mid \mathcal {N}_{E}\Psi =\Psi ,\,P\Psi =k\Psi \right\} $ and hence an isomorphism for dimensional reasons.

Similarly, we find as $\left.H_{\text {eff}}\right|_{\mathcal {N}_{E}=1}=2\sum _{l\in \mathbb {Z}_{\ast }^{3}}\sum _{p,q\in L_{l}}\left\langle e_{p},\widetilde {E}_{l}e_{q}\right\rangle b_{l,p}^{\ast }b_{l,q}$ that for any $\phi ,\varphi \in L^{2}(L_{k})$ ,

(10.55) $$ \begin{align} \left\langle U\phi,H_{\text{eff}}U\varphi\right\rangle & =2\sum_{l\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{l}}\left\langle e_{p},\widetilde{E}_{l}e_{q}\right\rangle \left\langle b_{l,p}b_{k}^{\ast}(\phi )\psi_{\mathrm{FS}},b_{l,q}b_{k}^{\ast}(\varphi )\psi_{\mathrm{ FS}}\right\rangle \\ & =2\sum_{k\in\mathbb{Z}_{\ast}^{3}}\sum_{p,q\in L_{k}}\left\langle e_{p},\widetilde{E}_{l}e_{q}\right\rangle \delta_{k,l}\left\langle \phi,e_{p}\right\rangle \left\langle e_{q},\varphi\right\rangle =2\left\langle \phi,\widetilde{E}_{k}\varphi\right\rangle, \nonumber \end{align} $$

whence $U^{\ast }H_{\mathrm {eff}}U=2\widetilde E_{k}$ . By elaborating the above argument slightly, one finds that the mapping

(10.56) $$ \begin{align} \tilde{U}:\bigoplus_{k\in\mathbb{Z}_{\ast}^{3}}L^{2}(L_{k})\rightarrow\left\{ \Psi\in\mathcal{H}_{N}\mid\mathcal{N}_{E}\Psi=\Psi\right\} \end{align} $$

defined by

(10.57) $$ \begin{align} \tilde{U}\bigoplus_{k\in\mathbb{Z}_{\ast}^{3}}\varphi_{k}=\sum_{k\in\mathbb{Z}_{\ast}^{3}}b_{k}^{\ast}\left(\varphi_{k}\right)\psi_{\mathrm{ FS}} \end{align} $$

is likewise a unitary isomorphism under which $\tilde {U}^{\ast }H_{\text {eff}}\tilde {U}=\bigoplus _{k\in \mathbb {Z}_{\ast }^{3}} \widetilde E_{k}$ .