A new second-order upper bound for the ground state energy of dilute Bose gases

Abstract

We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb {R}^3)$ , gives better rates and, in our opinion, is substantially simpler.

1 Introduction and main result

We consider N bosons in a finite box $\Lambda _L=[-\tfrac L 2, \tfrac L 2]^3 \subset \mathbb {R}^3$ , interacting via a two-body nonnegative, radial, compactly supported potential V with scattering length $\mathfrak {a}$ . The Hamilton operator has the form

(1.1) $$ \begin{align} H_L = -\sum_{i=1}^N \Delta_i+\sum_{1 \leq i<j \leq N}V(x_i-x_j) \end{align} $$

and acts on the Hilbert space $L^2_s(\Lambda _L^N)$ , the subspace of $L^2(\Lambda _L^N)$ consisting of functions that are symmetric with respect to permutations of the N particles (we use here units with particle mass $m=1/2$ and $\hbar =1$ ). We assume Dirichlet boundary conditions and denote by $E(N,L)$ the corresponding ground state energy. We are interested in the energy per unit volume in the thermodynamic limit, defined by

(1.2) $$ \begin{align} e(\rho)=\lim_{\substack{N,L\to+\infty\\ \rho=N/L^3}} \frac{E(N,L)}{L^3}. \end{align} $$

Bogoliubov [4] and, later, in more explicit terms, Lee–Huang–Yang [14] predicted that, in the dilute limit $\rho \mathfrak {a}^3 \ll 1$ , the specific ground state energy (1.2) is so that

(1.3) $$ \begin{align} e(\rho)= 4 \pi \mathfrak{a} \rho^2 \left[ 1+ \frac{128}{15\sqrt \pi} (\rho \mathfrak{a}^3)^{1/2}+ o( (\rho \mathfrak{a}^3)^{1/2}) \right]. \end{align} $$

In particular, up to lower order corrections, it only depends on the interaction potential through the scattering length $\mathfrak {a}$ . An alternative heuristic derivation of (1.3) was proposed in [15] (this approach was based on a partial differential equation with a convolution nonlinearity, whose mathematical properties were recently studied in [7]).

On the rigorous level, the validity of the leading term on the right-hand side of (1.3) was established by Dyson, who obtained an upper bound in [8], and by Lieb–Yngvason, who proved the matching lower bound in [16]. An upper bound with the correct second-order contribution was first derived in [20] by Yau–Yin for regular potentials, improving a previous estimate from [9], which only recovered the correct formula (as an upper bound) in the limit of weak coupling. The approach of [20] has been reviewed and adapted to a grand canonical setting in [1]. As for the lower bound, preliminary results have been obtained in [13] and [6], where (1.3) was shown in particular regimes, where the potential scales with the density $\rho $ . Finally, a rigorous lower bound matching (1.3) has been obtained by Fournais–Solovej, in [10] for $L^1$ potentials and, very recently, in [11] for a hard sphere interaction (a nonoptimal bound for hard spheres had been previously obtained in [5]).

Our goal in this article is to show a new upper bound for (1.3). With respect to the upper bound established in [20], our result holds for a larger class of potentials (in [20], the upper bound is proven for smooth potentials), it gives a better rate (although still far from optimal) and, most important in our opinion, it relies on a simpler proof.

Theorem 1.1. Let $V \in L^{3}(\mathbb {R}^3)$ be nonnegative, radially symmetric, with $\mathrm {supp}(V)\subset B_R(0)$ and scattering length $\mathfrak {a}$ . Then, the specific ground state energy $e(\rho )$ of the Hamilton operator $H_L$ defined in (1.1) satisfies

(1.4) $$ \begin{align} e(\rho)\leq 4\pi \rho^2 \mathfrak{a} \Big[ 1 + \frac{128}{15 \sqrt \pi} (\rho \mathfrak{a}^3)^{1/2} \Big] + C \rho^{5/2+ 1/10} \end{align} $$

for some $C>0$ (depending on $\| V \|_3$ and on R) and for $\rho $ small enough.

Remark. Since Dirichlet boundary conditions lead to the largest energy, the upper bound (1.4) holds in fact for arbitrary boundary conditions.

Remark. At the cost of a longer proof, we could improve the bound on the error, up to the order $\rho ^{5/2+2/9}$ (this is the rate determined by Lemma 5.1).

The proof of 1.4 is based on the construction of an appropriate trial state. However, we do not directly construct a trial state in $L^2_s (\Lambda _L^N)$ for the Hamiltonian (1.1). Instead, to simplify the analysis, it is very convenient to (1) consider smaller boxes (rather than letting $N, L \to \infty $ first and considering small $\rho $ at the end, we will consider a diagonal limit, with $L = \rho ^{-\gamma }$ , for some $\gamma> 1$ ), (2) work with periodic rather than Dirichlet boundary conditions and (3) work in the grand-canonical setting, considering states with variable number of particles, rather than the canonical setting. In other words, our trial state will be defined on the bosonic Fock space

$$ \begin{align*} \mathcal{F}(\Lambda_{ L})= \bigoplus_{n \geq 0} L^2_s (\Lambda_L^{n}) = \bigoplus_{n \geq 0} L^2 (\Lambda_L)^{\otimes_s n} \end{align*} $$

where $L^2_s (\Lambda _L^{n})$ is the subspace of $L^2 (\Lambda _L^n)$ consisting of wave functions that are symmetric with respect to permutations. On $\mathcal {F}(\Lambda _{ L})$ , we consider the number of particles operator $\mathcal {N}$ defined through $(\mathcal {N} \psi )^{(n)}= n \psi ^{(n)}$ . Moreover, we introduce the Hamiltonian operator $\mathcal {H}$ , setting

(1.5) $$ \begin{align} (\mathcal{H} \psi)^{(n)}= \mathcal{H}^{(n)} \psi^{(n)} \end{align} $$

with

$$ \begin{align*} \mathcal{H}^{(n)} = \sum_{j=1}^n -\Delta_{x_j} + \sum_{1 \leq i<j \leq n} V(x_i - x_j), \end{align*} $$

imposing now (in contrast to what we did in (1.1)) periodic boundary conditions (with a slight abuse of notation, V denotes here the periodic extension of the potential introduced in (1.1)). The upper bound for the energy of (1.5) will then imply Theorem 1.1 thanks to the following localisation result.

Proposition 1.2. Let $e(\rho )$ be defined as in (1.2), with Dirichlet boundary conditions. Let $R < b < L$ , with R the radius of the support of the potential V, as defined in Theorem 1.1. Then, for any normalised $\Psi _{L} \in \mathcal {F} (\Lambda _{L})$ satisfying periodic boundary conditions and such that

(1.6) $$ \begin{align} \langle \Psi_{L}, \mathcal{N} \Psi_{L} \rangle \geq \rho(1+c' \rho) (L + 2 b + R)^3 ,\quad \langle \Psi_{L}, \mathcal{N}^2 \Psi_{L} \rangle \leq C' \rho^{2} (L + 2 b + R)^6 \end{align} $$

for some $c' , C'> 0$ , we have

(1.7) $$ \begin{align} e(\rho) \leq \frac{\langle \Psi_{L}, \mathcal{H} \Psi_{L} \rangle }{L^3} + \frac{C}{L^{4} b} \langle \Psi_{L}, \mathcal{N} \Psi_{L} \rangle \end{align} $$

for a universal constant $C> 0$ .

The proof of Proposition 1.2 is standard; see [18, 20, 1]. Roughly speaking, the idea consists in using $\Psi _L$ (satisfying periodic boundary conditions on the box $\Lambda _L$ ) to construct a trial state satisfying Dirichlet boundary conditions on a slightly larger box of side length $L + 2b$ and then in approaching the thermodynamic limit by replicating the Dirichlet state on several boxes of side length $L+2b$ , separated by corridors of size R (to avoid interactions among different boxes). For completeness, we provide a detailed proof of Proposition 1.2 in Appendix A.

The bulk of the article contains the proof of the following proposition, establishing the existence of a trial state with the correct energy per unit volume and the correct expected number of particles on boxes of size $L = {\tilde {\rho }}^{-\gamma }$ . We use here the notation ${\tilde {\rho }}$ for the density to stress the fact that the upper bound (1.9) will be inserted in (1.7) to prove an upper bound for the specific ground state energy $e (\rho )$ , for a slightly different density $\rho < {\tilde {\rho }}$ (to make up for the corrections on the right-hand side of (1.6)).

Proposition 1.3. As in Theorem 1.1, assume that $V \in L^3 (\mathbb {R}^3)$ is nonnegative, radially symmetric with $\text {supp } V \subset B_R (0)$ and scattering length $\frak {a}$ . For $\gamma> 1$ and ${\tilde {\rho }}> 0$ let $L = {\tilde {\rho }}^{-\gamma }$ . Then, for every $0< \varepsilon < 1/4$ , there exists $\Phi _{{\tilde {\rho }}} \in \mathcal {F} (\Lambda _L)$ satisfying periodic boundary conditions such that

(1.8) $$ \begin{align} \langle \Phi_{{\tilde{\rho}}} , \mathcal{N} \Phi_{{\tilde{\rho}}} \rangle \geq {\tilde{\rho}} L^3 , \qquad \langle \Phi_{{\tilde{\rho}}} , \mathcal{N}^2 \Phi_{{\tilde{\rho}}} \rangle \leq C {\tilde{\rho}}^{2} L^6 \end{align} $$

and

(1.9) $$ \begin{align} \frac{{\bigl\langle\Phi_{{\tilde{\rho}}},\mathcal{H} \Phi_{{\tilde{\rho}}}\bigr\rangle}}{ L^3} \leq 4\pi\mathfrak{a} {\tilde{\rho}}^{\,2}\bigg(1+\frac{128}{15\sqrt{\pi}}(\mathfrak{a}^3 {\tilde{\rho}})^{1/2}\bigg)+ \mathcal{E} , \end{align} $$

with

$$ \begin{align*} \mathcal{E} \leq C \,{\tilde{\rho}}^{\,5/2} \cdot \max \{{\tilde{\rho}}^\varepsilon , \, {\tilde{\rho}}^{4-3\gamma -6\varepsilon} , \, {\tilde{\rho}}^{9/4-3\gamma/2-3\varepsilon} \}. \end{align*} $$

Remark. the condition $\gamma> 1$ is needed to make sure that the localisation error in (1.7) is negligible. While we will choose $\gamma = 11/10$ to optimise the rate, our analysis allows us to take any $1 < \gamma < 4/3$ . With a longer proof, our techniques could be extended to all $1 < \gamma < 5/3$ . This suggests that our trial state captures the correct correlations of the ground state, up to length scales of the order $\rho ^{-5/3}$ .

With Proposition 1.2 and Proposition 1.3 we can prove Theorem 1.1.

Proof Proof of Theorem 1.1.

For given $\rho> 0$ , we would like to choose ${\tilde {\rho }}$ or, equivalently, $L = {\tilde {\rho }}^{-\gamma }$ , so that (1.8) implies (1.6). Fixing $c'> 0$ and $b = L^\alpha $ , for some $\alpha \in (0;1)$ , this leads to the implicit equation

(1.10) $$ \begin{align} L = {\tilde{\rho}}^{-\gamma} = \left[ \rho (1+ c' \rho) (1 + 2 L^{\alpha-1} + R L^{-1})^3 \right]^{-\gamma}. \end{align} $$

Setting $L = \big (\rho (1+c' \rho ) \big )^{-\gamma } x$ , we rewrite (1.10) as

$$ \begin{align*} x = \big(1 + 2 \, \big(\rho(1+c' \rho) \big)^{\gamma (1-\alpha)} / x^{1-\alpha} + R \, \big(\rho(1+c' \rho) \big)^\gamma /x \big)^{-3\gamma} \end{align*} $$

and we conclude that the existence of a solution $L = L(\rho )$ of (1.10) follows from the implicit function theorem, if $\rho> 0$ is small enough (the solution stems from $x=1$ for $\rho = 0$ ). By construction, $L = {\tilde {\rho }}^{-\gamma }$ , with

$$ \begin{align*} {\tilde{\rho}} = \rho (1+c' \rho) (1 + 2 {\tilde{\rho}}^{\gamma (1-\alpha)} + R {\tilde{\rho}}^\gamma)^3\end{align*} $$

and thus

(1.11) $$ \begin{align} \rho \leq {\tilde{\rho}} \leq \rho (1 + C \rho + C \rho^{\gamma (1-\alpha)}) . \end{align} $$

From Proposition 1.3, we find $\Phi _{{\tilde {\rho }}} \in \mathcal {F} (\Lambda _L)$ such that (1.8) and (1.9) hold true. In particular, (1.8) implies (1.6) (with $b = L^\alpha $ , $C'=C$ ). Thus, from Proposition 1.2 we conclude

$$ \begin{align*} e(\rho) \leq \frac{\langle \Phi_{{\tilde{\rho}}}, \mathcal{H} \Phi_{{\tilde{\rho}}} \rangle}{L^3} + \frac{C}{L^4 b} \langle \Phi_{{\tilde{\rho}}} , \mathcal{N} \Phi_{{\tilde{\rho}}} \rangle. \end{align*} $$

Inserting (1.9) and (1.8), we obtain (since (1.8) also implies that $\langle \Phi _{{\tilde {\rho }}} , \mathcal {N} \Phi _{{\tilde {\rho }}} \rangle \leq C \widetilde {\rho } L^3$ )

$$ \begin{align*} e (\rho) \leq 4 \pi \frak{a} {\tilde{\rho}}^2 \left[ 1 + \frac{128}{15\sqrt{\pi}} (\frak{a}^3 {\tilde{\rho}})^{1/2} \right] + C {\tilde{\rho}}^{1+\gamma (1+\alpha)} + C {\tilde{\rho}}^{\,5/2} \cdot \max \{{\tilde{\rho}}^\varepsilon , \, {\tilde{\rho}}^{4-3\gamma -6\varepsilon} , \, {\tilde{\rho}}^{9/4-3\gamma/2-3\varepsilon} \}. \end{align*} $$

With (1.11), we conclude that

$$ \begin{align*} e (\rho) \leq \; &4 \pi \frak{a} \rho^2 \left[ 1 + \frac{128}{15\sqrt{\pi}} (\frak{a}^3 \rho )^{1/2} \right] \\ &+ C \rho^{5/2} \cdot \max \{ \rho^{\gamma (1-\alpha)-1/2} , \, \rho^{\gamma (1+\alpha) -3/2} , \, \rho^\varepsilon , \, \rho^{4-3\gamma -6\varepsilon} , \, \rho^{9/4-3\gamma/2-3\varepsilon} \} ,\end{align*} $$

where we neglected errors of order $C \rho ^3$ , which are subleading compared with $C\rho^{5/2+\varepsilon}$ , since $\varepsilon \in (0;1/4)$ .

Comparing the first two errors, we choose $\alpha = 1/(2\gamma )$ . Comparing instead third and fourth errors, we set $\varepsilon = (4-3\gamma )/7$ (both choices are consistent with the conditions $\alpha \in (0;1)$ and $\varepsilon \in (0;1/4)$ , because $\gamma> 1$ ). Since, with these choices, the last error is of smaller order, we obtain

$$ \begin{align*} e (\rho) \leq \; 4 \pi \frak{a} \rho^2 \left[ 1 + \frac{128}{15\sqrt{\pi}} (\frak{a}^3 \rho )^{1/2} \right] + C \rho^{5/2} \cdot \max \{ \rho^{\gamma-1}, \rho^{(4-3\gamma)/7} \}. \end{align*} $$

Choosing $\gamma = 11/10$ , we find (1.4).

The proof of Proposition 1.3 occupies the rest of the article (excluding Appendix A, where we show Proposition 1.2). In Section 2 we define our trial state. To this end, we will start with a coherent state describing the Bose–Einstein condensate. Similar to [12, 9], we will then act on the coherent state with a Bogoliubov transformation to add the expected correlation structure. Finally, we will apply the exponential of a cubic expression in creation operators. While the Bogoliubov transformation creates pairs of excitations with opposite momenta $p, -p$ , the cubic operator creates three excitations at a time, two with large momenta $r+v, -r$ and one with low momentum v. This last step is essential, since, as follows from [9, 17], quasi-free states cannot approximate the ground state energy to the precision of (1.3). We remark that the idea of creating triples of excitations originally appeared in the work of Yau–Yin [20] (a brief comparison with the trial state of [20] can be found after the precise definition of our trial state in (2.25)). Recently, it has been also applied to establish the validity of Bogoliubov theory in the Gross–Pitaevskii regime in [3, 2]; while our approach is inspired by these papers, we need new tools to deal with the large boxes considered in Proposition 1.3 (a simple computation shows that the Gross–Pitaevskii regime corresponds to the exponent $\gamma = 1/2$ ; to control localisation errors, we need instead to choose $\gamma> 1$ ). In Section 3, we combine the contributions to the energy of the trial state arising from the conjugation with the Bogoliubov transformation and from the action of the cubic phase, proving the desired upper bound. In Section 4 and Section 5, we prove technical bounds which allow us to identify the leading contributions collected in Section 3.

2 Setting and trial state

To show Proposition 1.3, we find it convenient to work with rescaled variables. We consider the transformation $x_j \to x_j / L$ and, motivated by the choice $L = {\tilde {\rho }}^{-\gamma }$ in Proposition 1.3, we set $N = {\tilde {\rho }}^{1-3\gamma }$ (we will look for trial states with expected number of particles close to N to make sure that (1.8) holds true). It follows that the Hamiltonian (1.5) is unitarily equivalent to the operator $L^{-2} \mathcal {H}_N = {\tilde {\rho }}^{2\gamma } \mathcal {H}_N$ , with $\mathcal {H}_N$ acting on the Fock space $\mathcal {F} (\Lambda )$ defined over the unit box $\Lambda = \Lambda _1 = [-1/2; 1/2 ]^3$ (with periodic boundary conditions) so that $(\mathcal {H}_N \Psi )^{(n)} = \mathcal {H}_N^{(n)} \Psi ^{(n)}$ , with

$$ \begin{align*} \mathcal{H}_N^{(n)} = \sum_{j=1}^n -\Delta_{x_j} + \sum_{1 \leq i,j \leq n} N^{2-2\kappa} V (N^{1-\kappa} (x_i - x_j)) \end{align*} $$

and $\kappa = (2\gamma - 1)/ (3\gamma -1)$ . The assumption $\gamma> 1$ in Proposition 1.3 allows us to restrict our attention to $\kappa \in (1/2; 2/3)$ .

For any momentum $p \in \Lambda ^* = 2\pi \mathbb {Z}^3$ , we introduce on the Fock space $\mathcal {F} (\Lambda ) = \bigoplus _{n \geq 0} L^2_s (\Lambda ^{n})$ the operators $a_p^*, a_p$ , creating and, respectively, annihilating a particle with momentum p. Creation and annihilation operators satisfy the canonical commutation relations

(2.1) $$ \begin{align} \left[ a_p , a_q^* \right] = \delta_{pq} , \qquad \left[ a_p , a_q \right] = \left[ a^*_p, a_q^* \right] = 0. \end{align} $$

On $\mathcal {F}(\Lambda )$ , we define the number of particles operator $\mathcal {N} = \sum _{p \in \Lambda ^*} a_p^* a_p$ . Expressed in terms of creation and annihilation operators, the Hamiltonian $\mathcal {H}_N$ takes the form

(2.2) $$ \begin{align} \mathcal{H}_N=\sum_{p\in\Lambda^*}p^2a_p^*a_p+\frac{1}{2N^{1-\kappa}} \sum_{p,q,r\in \Lambda^*} \widehat{V} (r/N^{1-\kappa}) \, a_{p+r}^*a_q^*a_{q+r}a_{p}. \end{align} $$

We now construct our trial state. To generate a condensate, we use a Weyl operator

(2.3) $$ \begin{align} W_{N_0}=\exp \big[ \sqrt{N_0} a_0^* - \sqrt{N_0}a_0 \big] \end{align} $$

with a parameter $N_0$ to be specified later on. While $W_{N_0}$ leaves $a_p, a_p^*$ invariant, for all $p \in \Lambda ^* \backslash \{ 0 \}$ , it produces shifts of $a_0, a_0^*$ ; in other words,

(2.4) $$ \begin{align} W_{N_0}^* a_0 \, W_{N_0} = a_0 + \sqrt{N_0} , \qquad W_{N_0}^* a_0^* \, W_{N_0} = a_0^* + \sqrt{N_0}. \end{align} $$

When acting on the vacuum vector $\Omega = \{ 1, 0, \dots \}$ , (2.3) generates a coherent state in the zero-momentum mode $\varphi _0 (x) \equiv 1$ , with expected number of particles $N_0$ .

It turns out, however, that the coherent state does not approximate the ground state energy, not even to leading order. To get closer to the ground state energy, it is crucial to add correlations among particles. To this end, we fix $0 < \ell < 1/2$ and we consider the lowest energy solution $f_{\ell }$ of the Neumann problem

(2.5) $$ \begin{align} \left[ -\Delta + \frac{1}{2} V \right] f_{\ell} = \lambda_{\ell} f_{\ell} \end{align} $$

on the ball $|x| \leq N^{1-\kappa }\ell $ , with the normalisation $f_\ell (x) = 1$ if $|x| =N^{1-\kappa } \ell $ . Furthermore, by rescaling, we define $f_{N,\ell } (x) := f_\ell \big ( N^{1-\kappa }x\big )$ for $|x| \leq \ell $ . We extend $f_{N,\ell }$ to a function on $\Lambda $ , by fixing $f_{N,\ell } (x) = 1$ , for all $x \in \Lambda $ , with $|x|> \ell $ . Then

(2.6) $$ \begin{align} \left[ -\Delta + \frac12 N^{2-2\kappa} V ( N^{1-\kappa}x ) \right] f_{N,\ell} (x) = N^{2-2\kappa} \lambda_\ell f_{N,\ell} (x) \chi_\ell(x) \end{align} $$

for all $x \in \Lambda $ , where $\chi _\ell $ denotes the characteristic function of the ball of radius $\ell $ . We denote by $ \widehat {f}_{N,\ell } (p)$ the Fourier coefficients of the function $f_{N,\ell }$ , for $p \in \Lambda ^*$ . We also define $w_\ell (x) = 1 - f_\ell (x)$ (with $w_\ell (x) = 0$ for $|x|> N^{1-\kappa } \ell $ ) and its rescaled version $w_{N,\ell } : \Lambda \to \mathbb {R}$ through $w_{N,\ell } (x) = w_\ell (N^{1-\kappa } x) = 1 - f_{N,\ell } (x)$ . The Fourier coefficients of $w_{N,\ell }$ are given by

$$ \begin{align*} \widehat{w}_{N,\ell} (p) = \int_{\Lambda} w_\ell ( N^{1-\kappa} x) e^{-i p \cdot x} dx =\frac1{N^{3-3\kappa}}\, \widehat{w}_\ell \big(p/N^{1-\kappa}\big) \end{align*} $$

where $\widehat {w}_\ell (k)$ denotes the Fourier transform of the (compactly supported) function $w_\ell $ . Some important properties of the solution of the eigenvalue problem (2.5) are summarised in the following lemma, whose proof can be found in [3, Appendix A] (replacing $N\in \mathbb {N}$ by $N^{1-\kappa }$ ).

Lemma 2.1. Let $V \in L^3 (\mathbb {R}^3)$ be nonnegative, compactly supported and spherically symmetric. Fix $\ell> 0$ and let $f_\ell $ denote the solution of (2.5). For $N\in \mathbb {N}$ large enough, the following properties hold true:

1. i) We have

   $$ \begin{align*} \bigg| \lambda_\ell - \frac{3\mathfrak{a} }{N^{3-3\kappa}\ell^3} \bigg| \leq \frac{1}{N^{3-3\kappa}\ell^3} \frac{C \mathfrak{a}^2}{\ell N^{1-\kappa}}. \end{align*} $$

2. ii) We have $0\leq f_\ell , w_\ell \leq 1$ . Moreover, there exists a constant $C> 0$ such that

   $$ \begin{align*} \left| \int V(x) f_\ell (x) dx - 8\pi \mathfrak{a} \right| \leq \frac{C \mathfrak{a}^2 }{\ell N^{1-\kappa}}. \end{align*} $$

3. iii) There exists a constant $C>0 $ such that, for all $x \in \mathbb {R}^3$ ,

   $$ \begin{align*} w_\ell(x)\leq \frac{C}{|x|+1} \quad\text{ and }\quad |\nabla w_\ell(x)|\leq \frac{C }{x^2+1}. \end{align*} $$

4. iv) There exists a constant $C> 0$ such that, for all $p \in \mathbb {R}^3$ ,

   $$ \begin{align*} |\widehat{w}_{N,\ell} (p)| \leq \frac{C}{ N^{1-\kappa}p^2} . \end{align*} $$

We consider the coefficients $\eta : \Lambda ^* \to \mathbb {R}$ defined through

(2.7) $$ \begin{align} \eta_p = -N \widehat{w}_{N,\ell} (p) = - \frac{N^\kappa}{N^{2-2\kappa}} \,\widehat{w}_\ell(p/N^{1-\kappa}). \end{align} $$

Lemma 2.1 implies that

(2.8) $$ \begin{align} |\eta_p| \leq \frac{C N^\kappa}{p^2} \end{align} $$

for all $p \in \Lambda _+^*=2\pi \mathbb {Z}^3 \backslash \{0\}$ and for some constant $C>0$ independent of $N\in \mathbb {N}$ (for $N\in \mathbb {N}$ large enough). From (2.6), we find the relation

(2.9) $$ \begin{align} p^2 \eta_p + \frac{N^\kappa}{2}\widehat{V} (p/N^{1-\kappa}) + \frac{1}{2N} \sum_{q \in \Lambda^*} N^\kappa\widehat{V} ((p-q) /N^{1-\kappa})\eta_q = N^{3-2\kappa} \lambda_{\ell} (\widehat{\chi}_\ell * \widehat{f}_{N,\ell}) (p). \end{align} $$

From Lemma 2.1, part iii), we also obtain

(2.10) $$ \begin{align} |\eta_0| \leq N^{3-2\kappa} \int_{\mathbb{R}^3} w_\ell (x) dx \leq C N^\kappa . \end{align} $$

The coefficients $\eta _p$ will be used to model, through a Bogoliubov transformation, short-distance correlations among particles. To reach this goal, it is enough to act on momenta $|p| \gg N^{\kappa /2}$ . On low momenta, the Bogoliubov transformation is needed to diagonalise the (renormalised) quadratic part of the Hamiltonian. For $\varepsilon> 0$ small enough, we therefore define the set

(2.11) $$ \begin{align} P_L \; = \Big\{ p \in \Lambda^*_+ : |p| \leq N^{\kappa/2+\varepsilon} \Big\}\, \end{align} $$

of low momenta. We will denote its complement by $P_L^c = \Lambda _+^* \backslash P_L$ . For $p \in \Lambda ^*_+$ we set

$$ \begin{align*} \nu_p = \tau_p \chi(p \in P_L) + \eta_p \chi(p \in P_L^c) \end{align*} $$

with $\eta _p$ defined in (2.7), $\tau _p \in \mathbb {R}$ defined by

(2.12) $$ \begin{align} \tanh(2\tau_p) = - \frac{8 \pi \mathfrak{a} N^\kappa}{p^2 +8 \pi \mathfrak{a} N^\kappa}\, \end{align} $$

and $\chi (p \in S)$ denoting the indicator function of the set S. With these coefficients, we define the Bogoliubov transformation

(2.13) $$ \begin{align} T_\nu \; = \exp \bigg(\;\frac 12 \sum_{p \in \Lambda^*_+} \nu_p \big(a^*_p a^*_{-p} - \mbox{h.c.} \big) \;\bigg). \end{align} $$

For any $p \neq 0$ we have

(2.14) $$ \begin{align} T^*_\nu a_p T_\nu = \gamma_p a_p + \sigma_p a^*_{-p} \end{align} $$

with the notation $\gamma _p = \cosh (\nu _p)$ and $\sigma _p = \sinh (\nu _p)$ .

With the Weyl operator (2.3) and the Bogoliubov transformation (2.13), we obtain the ‘squeezed’ coherent state $\widetilde {\Psi }_N = W_{N_0} T_\nu \Omega $ . Choosing $N_0$ so that $N = N_0 + \| \sigma \|^2$ , one can show that this trial state has approximately N particles and, to leading order, the correct ground state energy. However, as observed in [9] (for a similar trial state) and later in [17], the energy of the quasi-free state $\widetilde {\Psi }_N$ does not match the second-order correction in (1.3). To prove Proposition 1.3, we therefore need to modify the trial state. We do so by replacing the vacuum $\Omega $ in the definition of $\widetilde {\Psi }_N$ by the normalised Fock space vector $\xi _\nu / \| \xi _\nu \|$ , with $\xi _\nu = e^{A_\nu } \Omega $ and the cubic phase

(2.15) $$ \begin{align} \begin{aligned} A_\nu =\frac 1 {\sqrt {N}} \sum_{\substack{r\in P_H,v \in P_S:\\r+v\in P_H}} \nu_r \sinh (\nu_v) \,a^*_{r+v} a^*_{-r} a^*_{-v} \Theta_{r,v} = \frac 1 {\sqrt {N}} \sum_{\substack{r\in P_H,v \in P_S:\\r+v\in P_H}} \eta_r \sigma_v \,a^*_{r+v} a^*_{-r} a^*_{-v} \Theta_{r,v}. \end{aligned}\end{align} $$

Here, we introduced the momentum sets

(2.16) $$ \begin{align} P_H &= \{ p \in \Lambda_+^* : |p|> N^{1-\kappa-\varepsilon} \}, \nonumber\\ P_S &=\Big\{p\in \Lambda^*_+: N^{\kappa/2-\varepsilon} \leq |p|\leq N^{\kappa/2 + \varepsilon}\Big\}. \end{align} $$

Notice that $P_S \subset P_L$ . On the other hand, to make sure that $P_H \cap P_L = \emptyset $ , from now on we will require that $\varepsilon> 0$ is so small that $3\kappa - 2 + 4 \varepsilon < 0$ . Moreover, in (2.15) we included, for every $r \in P_H$ and every $v \in P_S$ , the cutoff

$$ \begin{align*} \Theta_{r,v}&= \prod_{s \in P_H}\!\Big[ 1-\chi(\mathcal{N}_s>0)\chi(\mathcal{N}_{-s+v}>0)\Big] \nonumber\\ & \quad\times \prod_{w \in P_S}\! \Big[ 1-\chi(\mathcal{N}_w>0)\chi(\mathcal{N}_{r-w}+\mathcal{N}_{-r-v-w}>0) \Big] \end{align*} $$

where $\mathcal {N}_p = a^*_p a_p$ and $\chi (t>0)$ is the indicator function of the set $\{ t>0 \}$ .

Remark. It is easy to check that the computation of the energy and the number of particles of the trial state we are constructing would not change substantially (and would still lead to a proof of Proposition 1.3), if in the definition (2.15) of $A_\nu $ we restricted the sum over r to the finite set $P_H \cap \{ p \in \Lambda ^*_+ : |p| < N^{1-\kappa + \varepsilon } \} = \{ p \in \Lambda ^*_+ : N^{1-\kappa -\varepsilon } < |p| < N^{1-\kappa +\varepsilon } \}$ . With this choice, the infinite product over $s \in P_H$ appearing in the definition of the cutoff $\Theta _{r,v}$ would be replaced by a finite multiplication.

Let us briefly discuss the action of the cutoff $\Theta _{r,v}$ . To understand its role in the computation of $e^{A_\nu } \Omega $ , we observe that, for every integer $m \geq 2$ , $r_1, \dots , r_m \in P_H$ , $v_1, \dots , v_m \in P_S$ , with $r_1 + v_1, \dots , r_m +v_m \in P_H$ , we find

$$ \begin{align*} \Theta_{r_m,v_m} &a^*_{r_{m-1}+v_{m-1}} a^*_{-r_{m-1}}a^*_{-v_{m-1}}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{-v_1}\Omega=\\ &= \prod_{i,j=1}^{m-1}\prod_{\substack{p_\ell\in\{-r_\ell,r_\ell+v_\ell\},\\\ell=i,j,m}}\delta_{p_i\neq -p_j+v_m}\delta_{-p_m+v_i\neq p_j} \\ &\hspace{4cm} \times a^*_{r_{m-1}+v_{m-1}}a^*_{-r_{m-1}}a^*_{-v_{m-1}}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{-v_1}\Omega. \end{align*} $$

The choice $i=j$ in the product on the second line introduces restrictions of the form $v_m \neq v_i$ and $p_m \neq p_i$ where $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}$ for $\ell = m,i$ , for all $i \in \{1, \dots , m-1 \}$ (the condition $p_i \neq -p_i + v_m$ , on the other hand, is trivially satisfied due to the assumption $p_i\in P_H,v_m\in P_S$ ). For $m\geq 3$ , the cutoff $\Theta _{r_m,v_m}$ implements additional restrictions involving three indices of the form $-p_i+v_j \neq p_k$ with $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}, \ell =i,j,k$ where $i,j,k = 1,\dots ,m, i\neq j\neq k$ , so that exactly one of the three indices is m. We conclude that, for any $m\geq 2$ ,

(2.17) $$ \begin{align} A_\nu^m\Omega = &\frac1{N^{m/2}}\sum_{\substack{r_1\in P_H,v_1\in P_S:\\r_1+v_1\in P_H}}\dots\sum_{\substack{r_m\in P_H,v_m\in P_S:\nonumber\\r_m+v_m\in P_H}}\prod_{i=1}^m \eta_{r_i}\sigma_{v_i}\nonumber\\ &\hspace{2cm} \times \theta(\{r_j,v_j\}_{j=1}^m)a^*_{r_m+v_m}a^*_{-r_m}a^*_{-v_m}\dots a^*_{r_1+v_1}a^*_{-r_1}a^*_{v_1}\Omega \end{align} $$

where

(2.18) $$ \begin{align} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) = \prod_{\substack{i,j,k=1\\j\neq k}}^m\prod_{\substack{p_i\in\{-r_i,r_i+v_i\}\\ p_k\in\{-r_k,r_k+v_k\}}}\delta_{-p_i+v_j\neq p_k}. \end{align} $$

To illustrate the reason for the introduction of the cutoff, let us compute the norm $\| \xi _\nu \|$ of the vector $\xi _\nu = e^{A_\nu } \Omega $ . With (2.17), we find

(2.19) $$ \begin{align} \| \xi_\nu\|^2 &= \sum_{m\geq 0}\frac{1}{(m!)^2} \| (A_\nu)^m \Omega \|^2 \nonumber\\ &= \sum_{m\geq 0}\frac1{(m!)^2}\frac1{N^m} \sum_{\substack{v_1, {\tilde{v}}_1 \in P_S \\ r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + {\tilde{v}}_1 \in P_H}} \cdots \sum_{\substack{v_m, {\tilde{v}}_m \in P_S \\ r_m, {\tilde{r}}_m \in P_H: \\ r_m + v_m ,\, {\tilde{r}}_m + {\tilde{v}}_m \in P_H}} \theta\big(\{{\tilde{r}}_j,{\tilde{v}}_j\}_{j=1}^m\big) \theta\big(\{r_j,v_j\}_{j=1}^m\big) \nonumber\\ & \hspace{1cm}\times\prod_{i=1}^m \eta_{r_i}\eta_{{\tilde{r}}_i} \sigma_{v_i} \sigma_{{\tilde{v}}_i} {\bigl\langle a^*_{r_m+v_m}a^*_{-r_m}a^*_{-v_m}\dots a^*_{-v_1}\Omega, a^*_{\tilde{r}_m+\tilde{v}_m}a^*_{-\tilde{r}_m}a^*_{-\tilde{v}_m} \dots a^*_{-\tilde{v}_1}\Omega\bigr\rangle}. \end{align} $$

Clearly, for the expectation on the last line not to vanish, all creation and annihilation operators with momenta in $P_S$ must be contracted among themselves. Since, on the support of $\theta (\{ r_j, v_j \}_{j=1}^m)$ , $v_i \neq v_j$ for all $i\neq j$ (and, similarly, ${\tilde {v}}_i \neq {\tilde {v}}_j$ for all $i\neq j$ on the support of $\theta (\{ \widetilde {r}_j, \widetilde {v}_j \}_{j=1}^m)$ ), we have $(m!)$ identical contributions arising from this pairing. We end up with

(2.20) $$ \begin{align} & \| \xi_\nu\|^2 =\sum_{m\geq 0}\frac1{m!}\frac1{N^m} \sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_m \in P_S, r_m, {\tilde{r}}_m \in P_H: \\ r_m + v_m ,\, {\tilde{r}}_m + v_m \in P_H}} \hskip -0.5cm \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{ {\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \nonumber\\ & \hspace{4.5cm}\times \prod_{i=1}^m \eta_{r_i}\eta_{{\tilde{r}}_i} \sigma^2_{v_i} {\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m} A^*_{\tilde{r}_m,v_m} \dots A^*_{\tilde{r}_1,v_1} \Omega\bigr\rangle} \end{align} $$

where we have introduced the notation $A_{r_i,v_i} = a_{r_i+v_i} a_{-r_i}$ .

It is now important to observe that, because of the presence of the cutoffs, the annihilation operators in $A_{r_j, v_j}$ must be contracted with the creation operators in $A_{\widetilde {r}_j, v_j}$ . In fact, if this was not the case, we would have $-r_j = -\widetilde {r}_\ell $ or $-r_j = \widetilde {r}_\ell + v_\ell $ and $r_j + v_j = -\widetilde {r}_k$ or $r_j + v_j = \widetilde {r}_k + v_k$ , with at least one of the two indices $\ell , k$ different from j. This would imply one of the four relations $\widetilde {r}_\ell + v_j = -\widetilde {r}_k$ , $\widetilde {r}_\ell + v_j = \widetilde {r}_k + v_k$ , $ \widetilde {r}_\ell + v_\ell = \widetilde {r}_k + v_j$ , $-\widetilde {r}_\ell - v_\ell + v_j = \widetilde {r}_k + v_k$ , all of which are forbidden by the cutoff $\theta \big ( \{ \widetilde {r}_j, \widetilde {v}_j \}_{j=1}^{m} \big )$ . We conclude that

(2.21) $$ \begin{align} {\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m} A^*_{\tilde{r}_m,v_m} \dots A^*_{\tilde{r}_1,v_1} \Omega\bigr\rangle} &\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{ {\tilde{r}}_j, {\tilde{v}}_j \}_{j=1}^{m} \big) \nonumber\\ &\quad = \prod_{i=1}^m \Big(\delta_{{\tilde{r}}_i, r_i} + \delta_{-{\tilde{r}}_i, r_i+v_i}\Big) \theta\big( \{r_j, v_j \}_{j=1}^{m} \big)\, \end{align} $$

(after identification of the momenta, the second cutoff becomes superfluous). From (2.20), we obtain

(2.22) $$ \begin{align} \|\xi_\nu \|^2 &= \sum_{m\geq 0}\frac1{m!}\frac1{N^{m}} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big(\{r_j, v_j\}_{j=1}^{m}\big) \, \prod_{i=1}^{m}\eta_{r_i} \big( \eta_{r_i} +\eta_{r_i+v_i} \big) \sigma_{v_i}^2\nonumber\\ &= \sum_{m\geq 0}\frac1{2^m m!}\frac1{N^{m}} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big(\{r_j, v_j\}_{j=1}^{m}\big) \, \prod_{i=1}^{m} \big( \eta_{r_i} +\eta_{r_i+v_i} \big)^2 \sigma_{v_i}^2 \end{align} $$

where we used the invariance of $\theta $ , with respect to $-r_i \to r_i + v_i$ . The cutoffs have been used first to exclude coinciding momenta in $v_1, \dots , v_m$ and in $\widetilde {v}_1, \dots , \widetilde {v}_m$ (which implies that, up to permutations, the pairing of the momenta in $P_S$ is unique) and then in (2.21) to make sure that annihilation operators in $A_{r_j, v_j}$ can only be contracted with the creation operators in $A^*_{\widetilde {r}_j, v_j}$ . This substantially simplifies computations. Similar simplifications will arise in the computation of the energy of our trial state.

Apart from the formula (2.22) for the norm $\| \xi _\nu \|^2$ , we will also need bounds on the expectation, in the state $\xi _\nu / \| \xi _\nu \|$ , of the number of particles operator $\mathcal {N}$ , of $\mathcal {N}^2$ , of the kinetic energy operator $\mathcal {K}$ and of the product $\mathcal {K} \mathcal {N}$ . These bounds are collected in the next proposition, whose proof will be discussed in Section 5.

Proposition 2.2. Let $\xi _\nu =e^{A_\nu }\Omega $ with $e^{A_\nu }$ defined in (2.15) with $\kappa \in (1/2 ;2/3)$ and $\varepsilon> 0$ such that $3\kappa -2 + 4\varepsilon < 0$ . Then, under the assumptions of Theorem 1.1, we have

(2.23) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{N}^{\,j} \xi_\nu \rangle}{\| \xi_\nu \|^2} & \leq C N^{(9\kappa/2-2+\varepsilon)j} \end{align} $$

and

(2.24) $$ \begin{align} \frac{ {\bigl\langle\xi_\nu,\mathcal{K} \mathcal{N}^{j-1} \xi_\nu\bigr\rangle}}{\|\xi_\nu\|^2} \leq C N^{5\kappa/2}N^{(9\kappa/2-2+\varepsilon)(j-1)} \end{align} $$

for $j=1,2$ .

Using the Weyl operator $W_{N_0}$ from (2.3), the Bogoliubov transformation $T_\nu $ defined in (2.13) and the cubic phase $A_\nu $ introduced in (2.15) (or, equivalently, the vector $\xi _\nu = e^{A_\nu } \Omega $ ), we can now define our trial state

(2.25) $$ \begin{align} \Psi_N = \frac{W_{N_0} T_\nu e^{A_\nu} \Omega}{\| W_{N_0} T_\nu e^{A_\nu} \Omega \|} = W_{N_0} T_\nu \frac{\xi_\nu}{\| \xi_\nu \|} . \end{align} $$

Here, we choose $N_0> 0$ such that

(2.26) $$ \begin{align} N = N_0 + \| \sigma_L \|^2 \end{align} $$

where $\sigma _L$ denotes the restriction to the set $P_L$ of the coefficients $\sigma _p = \sinh (\nu _p)$ , with $\nu _p$ defining the Bogoliubov transformation $T_\nu $ ; see (2.13).

Let us briefly compare our trial state with the one of [20]. In both approaches, the condensate is perturbed with operators creating double and triple excitations, the latter having two particles with high momenta and one particle with low momentum. Moreover, similarly as in [20], we impose cutoffs making sure that each low momentum appears only once. In contrast to [20], we also impose cutoffs on high momenta. Moreover, we have a clearer separation between creation of pairs (obtained through the Bogoliubov transformation $T_\nu $ ) and creation of triples. Finally, in our approach, we create triple excitations through the action of $e^{A_\nu }$ on the vacuum; the algebraic structure of the exponential makes the analysis and the combinatorics much simpler.

As shown in the next proposition, the choice (2.26) of $N_0$ guarantees that $\Psi _N$ has the expected number of particles.

Proposition 2.3. Let $\Psi _N$ be defined in (2.25) with the parameter $N_0$ appearing in (2.3) defined by (2.26). Let $\kappa \in (1/2; 2/3)$ and $\varepsilon> 0$ so that $3\kappa -2 + 4\varepsilon < 0$ . Then

(2.27) $$ \begin{align} \langle \Psi_N, \mathcal{N} \Psi_N \rangle \geq N ,\qquad \langle \Psi_N, \mathcal{N}^2 \Psi_N \rangle \leq C N^2 \end{align} $$

for all N large enough.

To prove Proposition 2.3 (and later to show other properties of the trial state $\Psi _N$ ) in the next lemma we collect some bounds for norms of the coefficients appearing in the definition of $A_\nu $ in (2.15). We denote here by $\eta _L, \eta _{L^c}, \eta _S, \eta _H$ the restriction of $\eta : \Lambda ^* \to \mathbb {R}$ to the set $P_L, P_L^c, P_S$ and, respectively, $P_H$ . Similarly, we define $\gamma _L, \gamma _{L^c}, \gamma _H, \gamma _S$ and $\sigma _L, \sigma _{L^c}, \sigma _H, \sigma _S$ .

Lemma 2.4. We have

$$ \begin{align*} &\| \eta_{L^c} \|^2 \leq C N^{3 \kappa/2 -\varepsilon} , \quad && \| \eta_{L^c} \|^2_{H^1} \leq C N^{1+\kappa}, \quad && \|\eta_{L^c} \|_\infty \leq C N^{-2\varepsilon} \\[0.2cm] &\| \eta_H \|^2 \leq C N^{3\kappa-1 +\varepsilon} , \quad && \| \eta_H \|^2_{H^1} \leq C N^{1+\kappa} , \quad && \|\eta_H \|_\infty \leq C N^{3\kappa-2 +2\varepsilon} . \end{align*} $$

In particular, this implies that $\|\gamma _H\|_\infty , \|\sigma _H\|_\infty \leq C$ . Moreover, we have

$$ \begin{align*} \|\gamma_L \|^2_\infty , \|\sigma_L \|^2_\infty \leq C N^{\kappa/2} , \qquad \quad \| \gamma_L \sigma_L \|_1 \leq C N^{3\kappa/2 +\varepsilon} \end{align*} $$

and

$$ \begin{align*} \| \gamma_L \|^2 \leq C N^{3\kappa/2 + 3\varepsilon}, \qquad \| \sigma_L \|^2 \leq C N^{3\kappa/2} \, , \qquad \quad \| \sigma_L \|^2_{H^1} \leq C N^{5\kappa/2 +\varepsilon}. \end{align*} $$

Finally, we observe that

$$ \begin{align*} \| \sigma_S\|^2 \leq C N^{3\kappa/2}, \qquad \| \sigma_S\|^2_{H^1} \leq C N^{5\kappa/2+\varepsilon}, \qquad \| \gamma_S \|_\infty^2, \| \sigma_S \|^2_\infty \leq C N^\varepsilon . \end{align*} $$

Proof. The bounds for $\|\eta _{L^c}\|$ , $\| \eta _H\|$ , $\|\eta _{L^c}\|_\infty $ and $\| \eta _H\|_\infty $ follow from (2.8). On the other hand, with the notation $\check {\eta } (x) = - N w_\ell ( N^{1-\kappa } x)$ for the function on $\Lambda $ with Fourier coefficients $\eta _p$ , we find from Lemma 2.1, part iii),

$$ \begin{align*} \| \eta_{L^c} \|_{H^1}^2 \leq C \sum_{p \in \Lambda^*} p^2 |\eta_p|^2 \leq C \int |\nabla \check{\eta} (x)|^2 dx \leq C N^{1+ \kappa} \int_{\mathbb{R}^3} \frac{1}{(|x|^2+1)^2}\, dx \leq C N^{1+\kappa} . \end{align*} $$

To show bounds for $\sigma _L, \gamma _L$ we observe that, with (2.12) and $\gamma _p^2 = 1 + \sigma _p^2$ , we obtain

(2.28) $$ \begin{align} \sigma^2_p = \frac{p^2 + 8\pi \frak{a} N^\kappa - \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}}{2 \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}} , \qquad \sigma_p \gamma_p = \frac{-8\pi \frak{a} N^\kappa}{2 \sqrt{|p|^4 + 16 \pi \frak{a} N^\kappa p^2}}. \end{align} $$

Recalling that $P_L = \{ p \in \Lambda ^*_+ : |p| \leq N^{\kappa /2+\varepsilon } \}$ , we find

(2.29) $$ \begin{align} \| \sigma_L\|^2_\infty \leq \mathop{\mathrm{sup}}\limits_{p \in P_L : |p| \leq N^{\kappa/2}} C \frac{N^{\kappa/2}}{|p|} + \mathop{\mathrm{sup}}\limits_{p \in P_L : |p|\geq N^{\kappa/2}} C \frac{N^{2\kappa}}{|p|^4} \leq C N^{\kappa /2}. \end{align} $$

Moreover, by definition of $P_S$ we get

$$ \begin{align*} \|\sigma_S\|_\infty^2\leq \mathop{\mathrm{sup}}\limits_{N^{\kappa/2-\varepsilon}\leq|p|\leq N^{\kappa/2}} C \frac{N^{\kappa/2}}{|p|} + \mathop{\mathrm{sup}}\limits_{p \in P_L: |p|\geq N^{\kappa/2}} C \frac{N^{2\kappa}}{|p|^4} \leq C N^\varepsilon. \end{align*} $$

Using again $\gamma ^2_p= 1+ \sigma ^2_p$ , we find $\| \gamma _L\|_\infty \leq C N^{\kappa /2}$ and $\|\gamma _S\|_\infty \leq C N^\varepsilon $ . Similarly, we obtain

(2.30) $$ \begin{align} \| \sigma_L\|^2 \leq C \sum_{p \in P_L : |p| \leq N^{\kappa/2}} \frac{N^{\kappa/2}}{|p|} + C \sum_{p \in P_L : |p|> N^{\kappa/2}} \frac{N^{2\kappa}}{|p|^4} \leq C N^{3\kappa/2} \end{align} $$

and thus $\|\gamma _L\|^2 \leq C N^{3\kappa /2 + 3 \varepsilon }$ . Moreover, we have

(2.31) $$ \begin{align} \| \sigma_L\|^2_{H_1} \leq C \sum_{p \in P_L : |p| \leq N^{\kappa/2}} |p| N^{\kappa/2} + C \sum_{p \in P_L : |p| \geq N^{\kappa/2}} \frac{N^{2\kappa} }{p^2} \leq C N^{5\kappa/2 + \varepsilon}. \end{align} $$

The bound for $\| \sigma _L \gamma _L\|_1$ is proved similarly, using the expression for $\gamma _p \sigma _p$ in (2.28). Finally, we note that the estimates (2.30) and (2.31) do not improve when we consider the restriction of $\sigma _p$ to $P_S\subset P_L$ ; hence, $\| \sigma _S\|^2 \leq C N^{3\kappa /2}$ and $\| \sigma _S\|^2_{H^1} \leq C N^{5\kappa /2+\varepsilon }$ .

We can now return to the proof of Proposition 2.3.

Proof Proof of Proposition 2.3.

From (2.4) and (2.14) we get

(2.32) $$ \begin{align} T_\nu^* W_{N_0}^* \mathcal{N} W_{N_0} T_\nu =\; &N_0+\sum_{p\in \Lambda_+^*}\sigma_p^2+\sqrt{N_0}(a_0+a^*_0)+ a^*_0 a_0 \nonumber\\ & + \sum_{p\in \Lambda^*_+}\big[ (\sigma_p^2+\gamma_p^2)a_p^*a_p + \gamma_p \sigma_p (a_p a_{-p} + \mbox{h.c.}) \big] . \end{align} $$

By definition of $\xi _\nu $ , $a_0 \xi _\nu = 0$ and $\langle \xi _\nu , a_p a_{-p} \xi _\nu \rangle = 0$ for every $p \in \Lambda ^*$ , as well as the definition $N = N_0 + \| \sigma _L \|^2$ , we obtain that

(2.33) $$ \begin{align} \langle \Psi_N, \mathcal{N} \Psi_N \rangle = \frac{\langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N} W_{N_0} T_\nu \xi_\nu \rangle}{\| \xi_\nu \|^2} = N +\sum_{p\in P_L^c}\sigma_p^2 + \sum_{p\in P_S\cup P_H} (\sigma_p^2+\gamma_p^2)\frac{\langle \xi_\nu,a_p^*a_p \xi_\nu\rangle}{\|\xi_\nu\|^2} . \end{align} $$

This immediately implies that $\langle \Psi _N, \mathcal {N} \Psi _N \rangle \geq N$ .

With $a_0 \xi _\nu = 0$ and the assumption $3\kappa - 2 + 4 \varepsilon < 0$ , (2.32) also implies that

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \; &C N^2 + C \sum_{p,q \in \Lambda^*_+} (\sigma_p^2 + \gamma_p^2) (\sigma_q^2 + \gamma_q^2) \langle \xi_\nu, a_p^* a_p a_q^* a_q \xi_\nu \rangle \\ &+ C \sum_{p,q \in \Lambda^*_+} \gamma_p \sigma_p \gamma_q \sigma_q \langle \xi_\nu, (a_p a_{-p} + a^*_p a_{-p}^*) (a_q a_{-q} + a_q^* a_{-q}^*) \xi_\nu \rangle. \end{align*} $$

Since $\xi _\nu $ is a superposition of states with $3m$ particles with momenta in $P_H \cup P_S$ , we obtain, writing $a_p a_{-p} a_q^* a_{-q}^* = a_q^* a_p a_{-q}^* a_{-p} +(\delta _{p,q}+ \delta _{-p, q})( a_{p}^* a_{p} +1) + \delta _{p,q} a^*_{-p} a_{-p}$ and similarly for $a_p^* a_{-p}^* a_q a_{-q}$ ,

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \; &CN^2 + C N^{2\varepsilon} \langle \xi_\nu , (\mathcal{N}+1)^2 \xi_\nu \rangle + C \sum_{p \in P_H \cup P_S} \gamma_p^2 \sigma_p^2 \langle \xi_\nu, a_p^* a_p \xi_\nu \rangle \\ & + C \sum_{p \in \Lambda^*_+} \gamma_p^2 \sigma_p^2 \| \xi_\nu \|^2 + C \hspace{-.3cm} \sum_{p,q \in P_H\cup P_S} \hspace{-.3cm} \gamma_p \sigma_p \gamma_q \sigma_q \langle \xi_\nu, a^*_p a_q a^*_{-p} a_{-q} \xi_\nu \rangle . \end{align*} $$

Estimating the last term through

$$ \begin{align*} \Big| \sum_{p,q \in P_H \cup P_S} \gamma_p &\sigma_p \gamma_q \sigma_q \langle \xi_\nu, a^*_p a_q a^*_{-p} a_{-q} \xi_\nu \rangle \Big| \\ \leq \; & \sum_{p,q \in P_H \cup P_S} |\gamma_p | | \sigma_p | |\gamma_q | | \sigma_q | \, \| a^* _q a_p \xi_\nu \| \| a^*_{-p} a_{-q} \xi_\nu \| \\ \leq \; & \sum_{p,q \in P_H \cup P_S} | \gamma_p | |\sigma_p | |\gamma_q | |\sigma_q | \, \big[ \| a_q a_p \xi_\nu \| + \| a_p \xi_\nu \| \big] \big[ \| a_{-p} a_{-q} \xi_\nu \| + \| a_{-q} \xi_\nu \| \big] \\ \leq \; & C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|_\infty^2 \| (\mathcal{N}+1) \xi_\nu \|^2 + C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|^2 \| \mathcal{N}^{1/2} \xi_\nu \|^2\\ &+ C \| \gamma_{S \cup H} \|_\infty^2 \| \sigma_{S \cup H} \|_\infty \| \sigma_{S \cup H} \| \| (\mathcal{N}+1) \xi_\nu \| \| \mathcal{N}^{1/2} \xi_\nu \|, \end{align*} $$

we conclude with the bounds in Lemma 2.4 and in Proposition 2.2 that, for $3\kappa -2 + 4\varepsilon < 0,$

$$ \begin{align*} \langle \xi_\nu , T_\nu^* W_{N_0}^* \mathcal{N}^2 W_{N_0} T_\nu \xi_\nu \rangle \leq \;& C N^2 + C N^{2\varepsilon} \langle \xi_\nu, \mathcal{N}^2 \xi_\nu \rangle + C N^\varepsilon \| \sigma \|^2 \langle \xi_\nu, \mathcal{N} \xi_\nu \rangle + C \| \gamma \|_\infty^2 \| \sigma \|^2 \\ \leq \; & C N^2 + C N^{9\kappa -4 + 4\varepsilon} + C N^{6\kappa -2 +2\varepsilon} + C N^{2\kappa} \leq C N^2. \\[-36pt] \end{align*} $$

The next theorem, whose proof occupies the rest of the article, determines the energy of $\Psi _N$ and, combined with Proposition 2.3, allows us to conclude the proof of Proposition 1.3.

Theorem 2.5. Let $\mathcal {H}_N$ and $\Psi _N \in \mathcal {F}$ be defined as in (2.2) and (2.25), respectively, and let $E_N^\Psi = \langle \Psi _N, \mathcal {H}_N \Psi _N \rangle $ . Let $\kappa \in (1/2; 2/3)$ , $\varepsilon> 0$ so small that $3\kappa -2 + 4\varepsilon < 0$ . Then, under the assumption of Theorem 1.1, we have

(2.34) $$ \begin{align} E_N^\Psi \leq 4 \pi \mathfrak{a} N^{1+\kappa} \bigg(1+\frac{128}{15\sqrt{\pi}}(\mathfrak{a}^3N^{3\kappa-2})^{1/2}\bigg)+ C N^{5\kappa/2} \max \{ N^{-\varepsilon} \hspace{-.1cm} , N^{9\kappa-5 +6\varepsilon} \hspace{-.1cm} ,N^{21\kappa/4-3+3\varepsilon}\} \end{align} $$

for all N large enough.

Remark. Equation (2.34) gives the correct second-order term for all $\kappa < 5/9$ (choosing $\varepsilon> 0$ small enough); this corresponds to exponents $\gamma < 4/3$ in Proposition 1.3. With a more complicated proof, we could have considered all $\kappa < 7/12$ (corresponding to $\gamma < 5/3$ ).

Proof Proof of Proposition 1.3.

Proposition 1.3 follows from Proposition 2.3 and Theorem 2.5, recalling that (1.5) is unitarily equivalent to $L^{-2} \mathcal {H}_N$ , with $\mathcal {H}_N$ as defined in (2.2), $L = {\tilde {\rho }}^{-\gamma }$ , $N = {\tilde {\rho }} L^3 = {\tilde {\rho }}^{1-3\gamma }$ and $\kappa = (2\gamma - 1)/ (3\gamma -1)$ . At the end, to obtain (1.8) and (1.9), we have to rename $\varepsilon (3\gamma - 1) \to \varepsilon $ .

3 Energy of the trial state

In this section we prove Theorem 2.5. With (2.25) and introducing the notations

(3.1) $$ \begin{align} \mathcal{G}_N = T^*_\nu \mathcal{L}_N T_\nu, \quad \text{with} \quad \mathcal{L}_N = W^*_{N_0} \mathcal{H}_N W_{N_0}\, \end{align} $$

we write

$$ \begin{align*} E^{\Psi}_N = \langle \Psi_N, \mathcal{H}_N \Psi_N \rangle = \frac{\langle \xi_\nu , \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2 } \end{align*} $$

with $\xi _\nu = e^{A_\nu } \Omega $ , as defined before (2.15). With (2.2) and recalling from (2.4) that

(3.2) $$ \begin{align} W^*_{N_0} a_p W_{N_0}=a_p+\sqrt{N_0}\,\delta_{p,0} \end{align} $$

we obtain $\mathcal {L}_N= \mathcal {L}^{(0)}_N +\mathcal {L}^{(1)}_N+ \mathcal {L}^{(2)}_N + \mathcal {L}^{(3)}_N + \mathcal {L}^{(4)}_N $ , with

(3.3) $$ \begin{align} \mathcal{L}^{(0)}_N=\;&\frac {N_0^2} {2} \,N^{\kappa-1} \widehat{V}(0)\nonumber\\ \mathcal{L}^{(1)}_N=\; & N_0^{3/2}\, N^{\kappa-1}\widehat{V}(0)(a_0+\mbox{h.c.})\nonumber\\ \mathcal{L}^{(2)}_N=\;&\sum_{p\in \Lambda^*}p^2a^*_pa_p + \frac{N_0}{N} \sum_{p\in \Lambda^*} N^\kappa \big( \widehat{V}(p/N^{1-\kappa}) + \widehat V(0) \big) a_p^*a_p \nonumber\\ & \hskip 3cm + \frac{N_0}{2 N}\sum_{p\in \Lambda^*}N^\kappa \widehat{V}(p/N^{1-\kappa})(a_p^*a_{-p}^*+ a_p a_{-p})\nonumber\\ \mathcal{L}^{(3)}_N=\; & \frac{\sqrt{N_0}}N \sum_{p,r\in \Lambda^*} N^{\kappa} \widehat{V}(r/N^{1-\kappa})(a^*_pa^*_ra_{p+r}+ a^*_{p+r} a_r a_p )\nonumber \\ \mathcal{L}^{(4)}_N=\; &\frac1{2N}\sum_{p,q,r \in \Lambda^*} N^{\kappa} \widehat{V}(r/N^{1-\kappa})a^*_{p+r} a^*_q a_{p}a_{q+r}. \end{align} $$

To compute $\mathcal {G}_N$ , we have to conjugate the operators in (3.3) with the Bogoliubov transformation $T_\nu $ . The result is described in the following proposition, whose proof will be discussed in Section 4.

Proposition 3.1. Let

(3.4) $$ \begin{align} \mathcal{K} = \sum_{p \in \Lambda^*_+} p^2 a^*_p a_p, \qquad \mathcal{V}_N^{(H)}= \frac{1}{2 N}\sum_{\substack{r\in \Lambda^*,\, p,q \in P_H:\\ p+r,q+r\in P_H}}N^\kappa \widehat{V}(r/N^{1-\kappa}) a_{p+r}^*a_q^*a_{p}a_{q+r} \, \end{align} $$

and (recalling from (2.26) that $N_0 = N - \| \sigma _L \|^2$ )

(3.5) $$ \begin{align} \mathcal{C}_N =\; \frac{\sqrt{N_0}}{ N} \sum_{\substack{p,r \in P_H\\ p+r \in P_S}} N^\kappa \widehat V(r/N^{1-\kappa})\, \sigma_{p+r} \gamma_p\gamma_r \,(a^*_{p+r} a^*_{-p} a^*_{-r} + \mbox{h.c.} ). \end{align} $$

Moreover, let

(3.6) $$ \begin{align} C_{\mathcal{G}_N} =\; & \frac{N^{1+\kappa}}{2} \widehat{V}(0) +\sum_{p\in {\Lambda}^*_+}p^2\sigma_p^2 + \sum_{p\in \Lambda^*_+} N^\kappa \widehat{V}(p/N^{1-\kappa})\sigma_p\gamma_p\nonumber\\ &+ \sum_{p\in P_L}N^\kappa \widehat{V}(p/N^{1-\kappa}) \sigma_p^2 +\frac{1}{2N}\sum_{\substack{p,r\in \Lambda^*_+\\r\neq p}} N^\kappa \widehat{V}(r/N^{1-\kappa})\sigma_p\sigma_{p-r}\gamma_p\gamma_{p-r} \nonumber\\ & - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p . \end{align} $$

Then, under the assumptions of Theorem 1.1, for all $\kappa \in (1/2; 2/3)$ and $\varepsilon> 0$ with $3\kappa -2 + 4\varepsilon < 0$ and N sufficiently large, we have

(3.7) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &C_{\mathcal{G}_N} + \frac{\langle \xi_\nu, (\mathcal{K}+\mathcal{V}_N^{(H)} +\mathcal{C}_N) \xi_\nu\rangle}{\| \xi_\nu \|^2}\nonumber \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6\varepsilon} , N^{21 \kappa /4 - 3 + 3 \varepsilon } \}. \end{align} $$

Remark. In fact, (3.7) does not only hold true for $\xi _\nu $ but for any state satisfying the bounds in Proposition 2.2.

The expectation of the operators $\mathcal {K}, \mathcal {V}_N^{(H)}, \mathcal {C}_N$ in the state $\xi _\nu / \| \xi _\nu \|$ , appearing on the right-hand side of (3.7) is determined by the next proposition, which will be shown in Section 5.

Proposition 3.2. Under the assumptions of Theorem 1.1, we have

$$ \begin{align*} \frac{\langle \xi_\nu, (\mathcal{K}+\mathcal{V}^{(H)}_N +\mathcal{C}_{N}) \xi_\nu \rangle }{\|\xi_\nu\|^2} \leq \; & \frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v}) \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} ,\, N^{12\kappa-7 +5\varepsilon}\} \end{align*} $$

for all $\kappa \in (1/2; 2/3)$ and all $\varepsilon> 0$ so small that $3\kappa -2 + 4\varepsilon < 0$ .

Let us now use the statements of Proposition 3.1 and Proposition 3.2 to obtain an upper bound for the energy of the trial state $\Psi _N$ and prove Theorem 2.5. From Proposition 3.1 and Proposition 3.2 we find

(3.8) $$ \begin{align} E_N^\Psi \leq \; & \frac{N^{1+\kappa}}{2}\,\widehat{V}(0)+ \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 + \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p \nonumber \\ & + \sum_{v \in P_L} N^\kappa \widehat V(v/N^{1-\kappa}) \sigma^2_v + \frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+\nonumber\\ p \neq q}} N^\kappa \widehat V( (p-q)/N^{1-\kappa}) \gamma_p \gamma_{q} \sigma_p \sigma_{q}\nonumber\\ & - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p \nonumber\\ & + \frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})\, + \mathcal{E} \end{align} $$

with

$$ \begin{align*} \mathcal{E} \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} \end{align*} $$

for all $\kappa \in (1/2; 2/3)$ and all $\varepsilon> 0$ with $3\kappa -2 + 4\varepsilon < 0$ (in this range, $12\kappa -7+5\varepsilon < 9\kappa -5+3\varepsilon $ ). Since $|\sigma _p - \eta _p| \leq C |\eta _p|^3$ for all $p \in P_L^c$ , with (2.8) we find

(3.9) $$ \begin{align} \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 \leq \sum_{v \in P_L} v^2 \sigma_v^2 + \sum_{p \in P_L^c} p^2 \eta_p^2 + C N^{5\kappa/2 - 3\varepsilon} . \end{align} $$

Similarly, with $|\gamma _p \sigma _p - \eta _p| \leq C \eta _p^3$ for all $p \in P_L^c $ , the last term on the first line of (3.8) can be written as

(3.10) $$ \begin{align} \sum_{p \in \Lambda^*_+} N^\kappa &\widehat V(p/N^{1-\kappa})\gamma_p \sigma_p \nonumber \\ & \leq \sum_{v \in P_L} N^\kappa \widehat V(v/N^{1-\kappa}) \gamma_v \sigma_v + \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p + C N^{5\kappa/2-3\varepsilon} . \end{align} $$

Next, we focus on the last term on the second line of (3.8). We define $\mathcal {E}_1$ through the identity

(3.11) $$ \begin{align} &\frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+\\ p \neq q}} N^\kappa \widehat V((p-q)/N^{1-\kappa}) \gamma_p \gamma_{q} \sigma_p \sigma_{q}\nonumber \\ &= \frac 1 {N} \!\!\sum_{\substack{p \in P_L^c, \\ q \in P_L}} N^\kappa \widehat V((p-q)/N^{1-\kappa}) \eta_p \gamma_{q} \sigma_{q} + \frac 1 {2N}\!\! \sum_{\substack{p,q \in P_L^c\\ p \neq q}} \!N^\kappa \widehat V((p-q)/N^{1-\kappa}) \eta_p \eta_q \!+ \mathcal{E}_1. \end{align} $$

Using again $|\gamma _p \sigma _p - \eta _p | \leq C |\eta _p|^3$ for all $p \in P_L^c$ , the estimate

(3.12) $$ \begin{align} \mathop{\mathrm{sup}}\limits_{r\in \Lambda^*}\,\sum_{s \in \Lambda^*_+} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| \leq C N^{1+\kappa} \end{align} $$

and the bounds from Lemma 2.4, we conclude (using the condition $3\kappa -2 + 5\varepsilon < 0$ ) that

$$ \begin{align*} \mathcal{E}_1 &\leq C \left[ N^{\kappa} \| \eta_{L^c} \|_\infty \| \eta_{L^c} \|^2 + N^{\kappa-1} \| \eta_{L^c} \|_\infty \| \eta_{L^c} \|^2 \| \sigma_L \gamma_L \|_1 + N^{\kappa-1} \| \sigma_L \gamma_L \|_1^2 \right] \leq C N^{5\kappa/2-\varepsilon}. \end{align*} $$

To prove (3.12) we use (2.8) and we remark that

$$ \begin{align*} \sum_{s \in \Lambda^* : |s| \leq N^{1-\kappa}} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| \leq C N^{2\kappa} \sum_{s \in \Lambda^*_+: |s| \leq N^{1-\kappa}} |s|^{-2} \leq C N^{1+\kappa} \end{align*} $$

and that, rescaling variables (setting ${\tilde {r}} = r/ N^{1-\kappa }$ ) and using an integral approximation,

$$ \begin{align*} \sum_{s \in \Lambda^*_+ : |s|> N^{1-\kappa}} N^\kappa |\widehat V((r-s)/N^{1-\kappa})| |\eta_s| &\leq C N^{1+\kappa} \sum_{s \in \Lambda^* / N^{1-\kappa}: |s| > 1} N^{-3(1-\kappa)} \, |\widehat{V} ({\tilde{r}} - s)| |s|^{-2} \\ &\leq C N^{1+\kappa} \int_{|s| > 1} |\widehat{V} ({\tilde{r}} - s)| |s|^{-2} ds \leq C_q N^{1+\kappa} \| \widehat{V} \|_q \end{align*} $$

for any $q < 3$ . With the assumption $V \in L^{q'} (\mathbb {R}^3)$ , for some $q'> 3/2$ , (3.12) follows by the Hausdorff–Young inequality.

Finally, we remark that the terms in the last two lines of (3.8) can be combined, using (2.8) and the bound $\| \sigma _L \|^2 \leq C N^{3\kappa /2}$ from Lemma 2.4, into

(3.13) $$ \begin{align} - \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p& +\frac1N\sum_{v\in P_L}\sigma_v^2\sum_{r\in P_H} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})\nonumber\\ &\leq \frac 1 N \sum_{v \in P_L} \sigma^2_v \sum_{r \in P_L^c} N^\kappa \widehat V(r/N^{1-\kappa}) \eta_{r+v} + C N^{5\kappa/2 - \varepsilon}. \end{align} $$

Inserting (3.9), (3.10), (3.11) and (3.13) into (3.8), we obtain

(3.14) $$ \begin{align} E_N^\Psi \leq & \, \frac{N^{1+\kappa}}{2} \widehat{V} (0)+ \sum_{p \in P_L^c}\big[ p^2 \eta_p + N^\kappa \widehat V(p/N^{1-\kappa}) + \frac 1 {2N} \sum_{r \in P_L^c} N^\kappa \widehat V((r-p)/N^\kappa) \eta_r \big] \eta_p \nonumber\\ & + \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big)\Big( N^\kappa \widehat V(v/N^{1-\kappa}) +\frac 1 N \sum_{r \in P_L^c} N^\kappa \widehat V((r-v)/N^{1-\kappa})\eta_{r} \Big) \Big] \nonumber\\ & + C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align} $$

Let us now consider the first square bracket on the right-hand side of (3.14). Using the scattering equation (2.9) we obtain

(3.15) $$ \begin{align} &\sum_{p \in P_L^c} \Big[ p^2 \eta_p + N^\kappa \widehat V(p/N^{1-\kappa}) + \frac 1 {2N} \sum_{r \in P_L^c} N^\kappa \widehat V( (r-p)/N^{1-\kappa}) \eta_r \Big] \eta_p \nonumber\\ &\qquad\qquad= \frac 1 2 \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa}) \eta_p - \frac 1 {2N} \sum_{\substack{p \in P_L^c\\ v \in P_L} } N^\kappa \widehat V((p-v)/N^{1-\kappa}) \eta_v \eta_p+ \mathcal{E}_2 \end{align} $$

with

$$ \begin{align*} \mathcal{E}_2 =\; N^{3-2\kappa} \lambda_{\ell} \sum_{p \in P_L^c} \big( \widehat \chi_\ell \ast \widehat f_{N,\ell}\big)_p \eta_p - \frac 1 {2N} \sum_{p \in P_L^c} N^\kappa \widehat V(p/N^{1-\kappa})\eta_p \eta_0 . \end{align*} $$

Using $N^{3-3\kappa } \lambda _{\ell } \leq C$ (Lemma 2.1), $\| \widehat \chi _\ell \ast \widehat f_{N,\ell }\| = \| \chi _\ell f_{N,\ell } \| \leq C$ and $\|\eta _{L^c}\|^2 \leq C N^{3\kappa /2 -\varepsilon }$ (Lemma 2.4) in the first term, (2.10) and (3.12) in the second term, we find $\mathcal {E}_2 \leq C N^{5\kappa /2-\varepsilon }$ (using that $3\kappa -2 + 4\varepsilon < 0$ and $\kappa> 1/2$ ).

As for the second square bracket on the right-hand side of (3.14), we write

(3.16) $$ \begin{align} \sum_{v \in P_L} \big(\sigma^2_v + \gamma_v \sigma_v \big) \Big( N^\kappa \widehat V(v/N^{1-\kappa}) & +\frac 1 N \sum_{r \in P_L^c} N^\kappa \widehat V((r-v)/N^{1-\kappa})\eta_{r} \Big) \nonumber\\ &= \sum_{v \in P_L} \big(\sigma^2_v + \gamma_v \sigma_v \big) N^\kappa \big( \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v + \mathcal{E}_3 \end{align} $$

with

$$ \begin{align*} \mathcal{E}_3 =\; -\frac 1 N \hskip -0.2cm \sum_{v,\,r\in P_L}\hskip -0.2cm N^\kappa \widehat V((r-v)/N^{1-\kappa}) \big(\sigma^2_v + \gamma_v \sigma_v \big)\eta_{r} - \frac {\eta_0} N \sum_{v \in P_L} N^\kappa \widehat V_{N}(v/N^{1-\kappa}) \big(\sigma^2_v + \gamma_v \sigma_v \big) . \end{align*} $$

With (2.8), Lemma 2.4, $|\eta _0| \leq C N^\kappa $ and the assumption $3\kappa -2 + 5\varepsilon < 0$ , we obtain $\mathcal {E}_3 \leq N^{5\kappa /2-\varepsilon }$ .

Inserting (3.15) and (3.16) in (3.14) and completing sums over p on the right-hand side of (3.15), we arrive at

(3.17) $$ \begin{align} E_N^\Psi \leq \; & \frac{N}{2} \big( N^\kappa \widehat V_{N}(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_0 \nonumber \\ &+ \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big) \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v - \frac {1} 2 N^\kappa \big( \widehat V(\cdot/N^{1-\kappa}) \ast \widehat f_{N,\ell} \big)_v \eta_v \Big] \nonumber \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\}. \end{align} $$

Let us now introduce the notation $\hat {g}_p = ( N^\kappa \widehat V(\cdot /N^{1-\kappa }) \ast \widehat f_{N,\ell })_p$ . Notice that

(3.18) $$ \begin{align} \big|\, \widehat g_0 - 8 \pi \mathfrak{a} N^\kappa \,\big| \leq C N^{2\kappa -1}, \qquad \big|\, \widehat g_p - \widehat g_0 \, \big| \leq C |p| N^{2\kappa -1} . \end{align} $$

With the expression (2.28), we obtain

$$ \begin{align*} \sum_{v \in P_L} \Big[ v^2 \sigma_v^2 + \big(\sigma^2_v + \gamma_v \sigma_v \big) \widehat g_v \Big] & =\frac 12 \sum_{v \in P_L} \left[ -v^2 - \widehat g_v + \frac{v^4 + v^2 (8\pi \frak{a} N^\kappa + \widehat g_v)}{\sqrt{v^4 + 16 \pi \frak{a} N^\kappa v^2}} \right] \\ & = \frac 12 \sum_{v \in P_L} \left[ \sqrt{v^4 + 16 \pi \frak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa \right] + \mathcal{E}_{4} \end{align*} $$

where, with (3.18) and $3\kappa -2 + 4\varepsilon < 0$ , we find

$$ \begin{align*} \mathcal{E}_{4} &= \frac 12 \sum_{v\in P_L} \left[ 8\pi \frak{a} N^\kappa - \widehat g_v + \frac{\widehat g_v - 8\pi \frak{a} N^\kappa}{\sqrt{1 + 16 \pi \frak{a} N^\kappa / v^2}} \right] \\ &\leq C \sum_{v\in P_L} | 8\pi \frak{a} N^\kappa - \widehat g_v | \;\frac{|\sqrt{1 + 16 \pi \frak{a} N^\kappa / v^2} - 1 |}{\sqrt{1 + 16 \pi \frak{a} N^\kappa /v^2}} \\ &\leq C N^{2\kappa-1} \Big[ \sum_{|v| < N^{\kappa/2}} (1+ |v|) + \sum_{v \in P_L : |v|> N^{\kappa/2}} \frac{N^{\kappa/2}}{|v|^2} \Big] \leq C N^{5\kappa/2-\varepsilon} . \end{align*} $$

Moreover, from (3.17) and with the scattering equation (2.9), we obtain

$$ \begin{align*} - \frac{N^\kappa}{2} \sum_{v \in P_L} \widehat{g}_v \eta_v \leq \sum_{v \in P_L} \frac{(8\pi \frak{a} N^\kappa )^2}{4v^2} + C N^{5\kappa/2-\varepsilon}. \end{align*} $$

Thus,

$$ \begin{align*} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} + \frac 12 \sum_{v \in P_L} \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa + \frac{(8 \pi \mathfrak{a} N^\kappa )^2}{2v^2} \Big] \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align*} $$

With

$$ \begin{align*} \left| \sqrt{v^4 + 16 \pi \mathfrak{a} N^\kappa v^2} - v^2 - 8 \pi \mathfrak{a} N^\kappa + \frac{(8 \pi \mathfrak{a} N^\kappa )^2}{2v^2} \right| \leq C \frac{N^{2\kappa}}{|v|^4} \end{align*} $$

we can replace, up to an error of order $N^{5\kappa /2-\varepsilon }$ , the sum over $P_L$ with a sum over all $\Lambda ^*_+$ . With the rescaling $v \to N^{\kappa /2} v$ , we arrive at

(3.19) $$ \begin{align} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} + \frac{N^\kappa}{2} \sum_{v \in 2\pi N^{-\kappa/2} \mathbb{Z}^3} \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} v^2} - v^2 - 8 \pi \mathfrak{a} + \frac{(8 \pi \mathfrak{a} )^2}{2v^2} \Big] \nonumber \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\}. \end{align} $$

Recognising that (3.19) defines a Riemann sum and explicitly computing

$$ \begin{align*} \frac{1}{2 (2\pi)^3} \int dv \, \Big[ \sqrt{v^4 + 16 \pi \mathfrak{a} v^2} - v^2 - 8 \pi \mathfrak{a} + \frac{(8 \pi \mathfrak{a} )^2}{2v^2} \Big] = 4 \pi \frak{a} \cdot \frac{128}{15 \sqrt{\pi}} \frak{a}^{3/2} \end{align*} $$

we conclude that

$$ \begin{align*} E^\Psi_N \leq \; & 4 \pi \mathfrak{a} N^{1+\kappa} \cdot \left[ 1 + \frac{128}{15 \sqrt{\pi}} \, (\frak{a}^3 N^{3\kappa-2})^{1/2} \right] \\ &+ C N^{5\kappa/2} \max \{ N^{-\varepsilon},\,N^{9\kappa-5 +6\varepsilon} ,\, N^{21\kappa/4-3+3\varepsilon}\} . \end{align*} $$

To compare the Riemann sum in (3.19) with the integral, we first removed contributions arising from $|v| \leq N^{-\varepsilon }$ using that $|F(v)| \leq C / v^2$ , for small v, with the definition $F(v) = \sqrt {v^4 + 16 \pi \mathfrak {a} v^2} - v^2 - 8 \pi \mathfrak {a} + (8 \pi \mathfrak {a} )^2 / 2v^2$ . For $|v|> N^{-\varepsilon }$ , we use that $|\nabla F (v)| \leq C |v|^{-3} (1+v^2)^{-1}$ to compare the value of $F(q)$ with $F(v)$ , for all q in the cube of size $2\pi N^{-\kappa /2}$ centred at v.

4 Bogoliubov transformation

In this section, we show Proposition 3.1. From the definition (3.1) and from (3.3), we obtain (since $T_\nu $ does not act on the zero momentum mode and since $a_0 \xi _\nu = 0$ )

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} = \frac{N_0^2}{2} N^{\kappa - 1} \widehat{V} (0) + \sum_{j=2}^4 \frac{\langle \xi_\nu, \mathcal{G}_N^{(j)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \end{align*} $$

with $\mathcal {G}_N^{(j)} = T_\nu ^* \mathcal {L}_N^{(j)} T_\nu $ , for $j=2,3,4$ .

We start from the contribution of $\mathcal {G}_N^{(2)}$ . We write $\mathcal {L}_N^{(2)}= \mathcal {K} + \mathcal {L}_N^{(2,V)}$ with

$$ \begin{align*} \mathcal{L}_N^{(2,V)} =\frac{N_0}{N}\sum_{p\in \Lambda^*} N^\kappa \big( \hat{V}(p/N^{1-\kappa}) + \hat V(0) \big) a_p^*a_p + \frac{N_0}{2N}\sum_{p\in \Lambda^*}N^\kappa \hat{V}(p/N^{1-\kappa})(a_p^*a_{-p}^*+\mbox{h.c.} ). \end{align*} $$

Using (2.14) we get

$$ \begin{align*} T^*_{\nu} \mathcal{K} T_\nu - \left[ \mathcal{K} + \sum_{p \in \Lambda_+^*} p^2 \sigma_p^2 \right] = 2 \sum_{p\in \Lambda^*_+} p^2 \sigma_p^2 a^*_p a_p + \sum_{p \in \Lambda^*_+} p^2 \big[\gamma_p\sigma_p ( a^*_p a^*_{-p}+\mbox{h.c.}) \big] := E_1 + E_2. \end{align*} $$

From (2.15), $\langle \xi _\nu , \text {E}_2 \xi _\nu \rangle = 0$ ( $\xi _\nu $ is a superposition of states with $3m$ particles, for $m \in \mathbb {N}$ ). To bound the expectation of $E_1$ on $\xi _\nu $ we notice that $\langle \xi _\nu ,a^*_pa_p\xi _\nu \rangle =0$ if $p\in \Lambda ^*_+\backslash (P_S\cup P_H)$ . Moreover, proceeding as in (2.31), we have

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p \in P_S} ( p^2 \sigma_p^2) \leq \mathop{\mathrm{sup}}\limits_{p \in P_S: |p|\leq N^{\kappa/2}} N^{\kappa/2} |p| + \mathop{\mathrm{sup}}\limits_{p \in P_S : |p|> N^{\kappa/2}} \frac{N^{2\kappa}}{p^2} \leq C N^{\kappa} \end{align*} $$

while

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p \in P_H} (p^2 \sigma_p^2) \leq \mathop{\mathrm{sup}}\limits_{|p| \geq N^{1-\kappa-\varepsilon}} \frac{N^{2\kappa}}{p^2} \leq C N^{-2+4\kappa+2\varepsilon} \leq C N^\kappa \end{align*} $$

because, by assumption, $3\kappa -2+2\varepsilon < 0$ . Hence,

(4.1) $$ \begin{align} |{\bigl\langle\xi_\nu, E_{1}\xi_\nu\bigr\rangle}|\leq C N^\kappa \, \| \mathcal{N}^{1/2}\xi_\nu \|^2 . \end{align} $$

We now consider the contribution from $ \mathcal {L}_N^{(2,V)}$ . Using again $\langle \xi _\nu , a_p^* a_{-p}^* \xi _\nu \rangle = 0$ for all $p \in \Lambda _+^*$ and $\langle \xi _\nu , a_p^* a_{p} \xi _\nu \rangle = 0$ for all $p \in \Lambda ^*_+\backslash (P_S \cup P_H)$ , a straightforward computation shows that

$$ \begin{align*} &\frac{\langle \xi_\nu, T^*_\nu \mathcal{L}_N^{(2,V)} T_\nu \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &\hspace{2cm} = \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p + \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \big(\widehat V(0) +\widehat V(p/N^{1-\kappa}) \big) \sigma^2_p \\ & \hspace{2.3cm} + \frac{N_0}{N} \sum_{p\in P_S \cup P_H} N^\kappa \big[\widehat V(p/N^{\kappa-1})(\gamma_p+\sigma_p)^2 +\widehat V(0) (\gamma^2_p + \sigma^2_p) \big] \frac{\langle \xi_\nu, a^*_p a_p \xi_\nu \rangle}{\| \xi_\nu \|^2} . \end{align*} $$

With the bounds $\|\gamma _S \|^2_\infty $ , $\|\sigma _S\|^2_\infty $ , $\|\sigma _H\|_\infty ^2$ , $\|\gamma _H\|_\infty ^2 \leq C N^{\varepsilon }$ from Lemma 2.4, with (4.1) and with the estimate $\langle \xi _\nu , \mathcal {N} \xi _\nu \rangle \leq C N^{9\kappa /2 -2+\varepsilon } \| \xi _\nu \|^2$ from Proposition 2.2, we conclude that

(4.2) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}^{(2)}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &\frac{\langle \xi_\nu, \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} + \sum_{p \in \Lambda^*_+} p^2 \sigma_p^2 + \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \widehat V(p/N^{1-\kappa}) \gamma_p \sigma_p \nonumber\\ &+ \frac{N_0}{N} \sum_{p \in \Lambda^*_+} N^\kappa \big(\widehat V(0) +\widehat V(p/N^{1-\kappa}) \big) \sigma^2_p + C N^{5\kappa/2-\varepsilon} \end{align} $$

using again the condition $3\kappa - 2 +4\varepsilon < 0$ .

Next, we study the contribution of $\mathcal {G}_N^{(3)} = T_\nu ^* \mathcal {L}^{(3)}_N T_\nu $ , with $\mathcal {L}^{(3)}_N$ as in (3.3). Recall the operator $\mathcal {C}_N$ , defined in (3.5). Taking into account the fact that $\xi _\nu $ is a superposition of vectors with $2m$ particles with momenta in $P_H$ and m particles with momenta in $P_S$ , for $m \in \mathbb {N}$ , we obtain that

$$ \begin{align*} \langle \xi_\nu, \mathcal{G}_N^{(3)} \xi_\nu \rangle = \langle \xi_\nu , \mathcal{C}_N \xi_\nu \rangle + \sum_{j=1}^3 \left[ \langle \xi_\nu, \text{F}_j \xi_\nu \rangle + \text{h.c.} \right] \end{align*} $$

with

$$ \begin{align*} \text{F}_1 &= \frac{\sqrt{N_0}}{N} \sum_{p,r \in P_H : p+r \in P_S} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_{p+r} \sigma_p \sigma_r \, a^*_{p+r} a^*_{-p} a^*_{-r} \\ \text{F}_2 &= \frac{\sqrt{N_0}}{N} \sum_{p \in P_H , r \in P_S : p+r \in P_H} N^\kappa \left[ \widehat V(r/N^{1-\kappa}) + \widehat{V} (p / N^{1-\kappa}) \right] \gamma_{p+r} \sigma_p \sigma_r \, a^*_{p+r} a^*_{-p} a^*_{-r} \\ \text{F}_3 &= \frac{\sqrt{N_0}}{N} \sum_{p \in P_H, r \in P_S : p+r \in P_H} N^\kappa \left[ \widehat V(r/N^{1-\kappa}) + \widehat V(p /N^{1-\kappa}) \right] \sigma_{p+r} \gamma_p \gamma_r \, a_{-p-r} a_{p} a_{r} . \end{align*} $$

Using $\|a^*_{-r} (\mathcal {N}+1)^{1/2}\xi _\nu \|\leq \|a_{-r} (\mathcal {N}+1)^{1/2}\xi _\nu \| + \|(\mathcal {N}+1)^{1/2}\xi _\nu \|$ , we can bound

$$ \begin{align*} \langle \xi_\nu , \text{F}_1 \xi_\nu \rangle \leq \; &C N^{\kappa -1/2} \| \gamma_S \|_\infty \sum_{p,r \in P_H} |\sigma_r| |\sigma_p | \, \| a_{p+r} a_{-p} (\mathcal{N} + 1)^{-1/2} \xi_\nu \| \\ &\hspace{4cm} \times \left[ \| a_{-r} (\mathcal{N} + 1)^{1/2} \xi_\nu \| + \| (\mathcal{N} + 1)^{1/2} \xi_{\nu} \| \right] \\ \leq \; &C N^{\kappa -1/2} \| \gamma_S \|_\infty \| \sigma_H \|_\infty \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_S \|_\infty \| \sigma_H \|^2 \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2. \end{align*} $$

With Lemma 2.4 and Proposition 2.2, we obtain

$$ \begin{align*} \frac{\langle \xi_\nu , \text{F}_1 \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq CN^{37\kappa /4 - 4 +5\varepsilon/2} + C N^{17\kappa/2 - 7/2 + 5\varepsilon/2} \leq C N^{5\kappa/2} \cdot N^{21\kappa /4 - 7/2 + 3\varepsilon/2} \end{align*} $$

from the assumption that $3\kappa -2 +4\varepsilon < 0$ . Similarly, we find

$$ \begin{align*} \langle \xi_\nu , \text{F}_2 \xi_\nu \rangle \leq \; & C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \sigma_S \|_\infty \| \sigma_H \| \, \| (\mathcal{N}+ 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \sigma_S \| \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2 \\ \leq \; &C \big[ N^{37\kappa/4-4+3\varepsilon} + N^{31\kappa/4-3+3\varepsilon/2} \big] \| \xi_\nu \|^2 \leq C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon /2} \| \xi_\nu \|^2 \end{align*} $$

and also

$$ \begin{align*} \langle \xi_\nu , \text{F}_3 \xi_\nu \rangle \leq \; & C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \gamma_S \|_\infty \| \sigma_H \| \, \| (\mathcal{N}+ 1)^{1/2} \xi_\nu \| \| (\mathcal{N}+ 1) \xi_\nu \| \\ &+ C N^{\kappa -1/2} \| \gamma_H \|_\infty \| \gamma_S \| \| \sigma_H \| \, \| (\mathcal{N} + 1)^{1/2} \xi_\nu \|^2 \\ \leq \; &C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon} \| \xi_\nu \|^2 . \end{align*} $$

Summarising, we have

(4.3) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{G}_N^{(3)} \xi_\nu \rangle}{ \| \xi_\nu \|^2} \leq \frac{\langle \xi_\nu , \mathcal{C}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} + C N^{5\kappa/2} \cdot N^{21\kappa/4 - 3 + 3 \varepsilon} . \end{align} $$

Finally, let us consider $\mathcal {G}_N^{(4)} = T_\nu ^* \mathcal {L}_N^{(4)} T_\nu $ . We decompose $\langle \xi _\nu , \mathcal {G}_N^{(4)} \xi _\nu \rangle = \sum _{j=1}^3 \langle \xi _\nu , \text {G}_j \xi _\nu \rangle $ with

$$ \begin{align*} \text{G}_1 &= \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_q \gamma_{p+r} \gamma_{q+r} a^*_{p+r} a^*_q a_{p} a_{q+r} \\ \text{G}_2 & = \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ r \neq q,-p}} N^\kappa \widehat V(r/N^{1-\kappa}) \big( \gamma_{p+r} \sigma_q a^*_{p+r} a_{-q} + \sigma_{p+r} \gamma_q a_{-p-r} a^*_q \big) \\[-0.5cm] & \hskip5.5cm\times \big( \gamma_{p} \sigma_{q+r} a_{p}a^*_{-q-r} + \sigma_{p} \gamma_{q+r} a^*_{-p} a_{q+r}\big)\\[0.2cm] \text{G}_3 & = \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ r \neq q,-p}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p \sigma_q \sigma_{p+r} \sigma_{q+r} a_{p+r} a_{q} a^*_{p} a^*_{q+r}. \end{align*} $$

To estimate contributions from $\text {G}_3$ , we arrange terms in normal order. We find

$$ \begin{align*} \text{G}_3 = \; & \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p \sigma_q \sigma_{p+r} \sigma_{q+r} a^*_{p} a^*_{q+r} a_{p+r} a_{q} \\ &+ \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p \in \Lambda^*_+ \\ p \neq -r}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma_p^2 \sigma^2_{p+r} \big( a^*_p a_p + a^*_{p+r}a_{p+r}\big) \\ &+ \frac 1 {N} \sum_{\substack{p,q \in \Lambda^*_+ }} N^\kappa \widehat V(0) \sigma_p^2 \sigma^2_{q} a^*_p a_p \\ &+ \frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p \in \Lambda^*_+ :\\ p\not = -r}} N^\kappa \widehat V(r/N^{1-\kappa}) \sigma^2_p \sigma^2_{p+r} + \frac 1 {2N} \sum_{\substack{p,q \in \Lambda^*_+}} N^\kappa \widehat V(0) \sigma^2_p \sigma^2_{q} . \end{align*} $$

Since $a_p\,\xi _\nu =0$ if $p\in \Lambda ^*_+\backslash (P_S\cup P_H)$ and $\|\sigma _H\|_\infty \leq \|\sigma _S\|_\infty $ , we find, by Cauchy–Schwarz,

(4.4) $$ \begin{align} \langle \xi_\nu, \text{G}_3 \xi_\nu \rangle &\leq C N^{\kappa -1} \left[ \| \sigma_S \|_\infty^2 \| \sigma \|^2 \| (\mathcal{N}+1) \xi_\nu \|^2 + \| \sigma \|^4 \| \xi_\nu \|^2 \right] \nonumber\\ &\leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} , N^{9\kappa -5 + 3\varepsilon} \} \| \xi_\nu \|^2 \end{align} $$

using Proposition 2.2 and $3\kappa -2 + 4\varepsilon < 0$ . We proceed similarly for $\text {G}_2$ . Through normal ordering, we get

$$ \begin{align*} \text{G}_2 = \; &\frac 1 {N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r}\sigma_q \sigma_{q+r} a^*_{p+r} a^*_{-q-r} a_{p} a_{-q} \\ &+ \frac 1 {N} \sum_{\substack{r \in \Lambda^*,\, p,q \in \Lambda^*_+ \\ -r \neq q,p}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_{p+r} \gamma_{q+r}\sigma_p \sigma_{q} a^*_{p+r} a^*_{-p} a_{-q} a_{q+r} \\ &+ \frac 1 {N} \sum_{\substack{p,q \in \Lambda^*_+ }} N^\kappa \widehat V(0) \gamma_p^2 \sigma^2_q a^*_p a_p + \frac 1 {N} \sum_{\substack{r \in \Lambda^*,\,p \in \Lambda^*_+ }} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p^2 \sigma^2_{p+r} a^*_p a_p \\ &+ \frac 2 {N} \sum_{\substack{r \in \Lambda^*,\,p \in \Lambda^*_+ }} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \sigma_p \gamma_{p+r} \sigma_{p+r} a^*_p a_p \\ &+ \frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} . \end{align*} $$

Keeping the last contribution intact and estimating the term on the fourth line distinguishing the two cases $(p+r) \in P_S$ and $(p+r) \in P_H$ , we arrive at

$$ \begin{align*} \langle \xi_\nu , \text{G}_2 \xi_\nu \rangle \leq \; &\frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} \| \xi_\nu \|^2 \\ &+ C N^{\kappa -1} \| \gamma_S \|_\infty^2 \| \sigma \|^2 \| (\mathcal{N}+1) \xi_\nu \|^2 \\ &+ C N^{\kappa -1} \| \gamma_{S \cup H} \|_\infty \| \sigma_{S \cup H} \|_\infty \\ & \hspace{2cm} \times \left[ \| \gamma_S \sigma_S \|_1 + \| \gamma_H \|_\infty \mathop{\mathrm{sup}}\limits_p \sum_{r \in \Lambda^*} \widehat{V} (r/ N^{1-\kappa}) |\eta_{p+r}| \right] \| \mathcal{N}^{1/2} \xi_\nu \|^2 . \end{align*} $$

With the bounds in Lemma 2.4 and in Proposition 2.2 and with (3.12), we conclude that

(4.5) $$ \begin{align} \langle \xi_\nu , \text{G}_2 \xi_\nu \rangle \leq \frac 1 {2N} \sum_{\substack{p,r \in \Lambda^*_+}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_{p+r} \sigma_p \sigma_{p+r} \| \xi_\nu \|^2 + C N^{5\kappa /2} \cdot N^{9\kappa -5 + 3\varepsilon} \| \xi_\nu \|^2. \end{align} $$

Finally, we consider $\text {G}_1$ . Recalling that $a_p \xi _\nu = 0$ if $p \in \Lambda ^*_+ \backslash (P_S \cup P_H)$ and observing that $\langle \xi _\nu , a^*_{p+r}a^*_qa_pa_{q+r}\xi _\nu \rangle \neq 0$ only if the operator $a^*_{p+r}a^*_q a_p a_{q+r}$ preserves the number of particles in $P_S$ and in $P_H$ , we arrive at

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \; &\frac 1 {2N} \sum_{\substack{r \in \Lambda^*,\, p,q \in P_H: \\p+r,q+r\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa}) \gamma_p \gamma_q \gamma_{p+r} \gamma_{q-r} \langle \xi_\nu , a^*_{p+r} a^*_q a_{p} a_{q+r} \xi_\nu \rangle \\ &+ C N^{\kappa -1 } \| \gamma_{S \cup H} \|_\infty^2 \| \gamma_S \|^2 \| (\mathcal{N} + 1) \xi \|^2. \end{align*} $$

With $|\gamma _p \gamma _q \gamma _{p+r} \gamma _{q+r} - 1| \leq C \| \eta _H \|_\infty ^2$ for all $p,q \in P_H$ , with $(p+r), (q+r) \in P_H$ and using the estimate (see the proof of (3.12))

$$ \begin{align*} \mathop{\mathrm{sup}}\limits_{p\in \Lambda^*} \sum_{r \in \Lambda^*_+ : r \not = p} \frac{N^\kappa |\widehat{V} (r/N^{1-\kappa})|}{|p-r|^2} \leq C N \end{align*} $$

we conclude that

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle + C \| \eta_H \|_\infty^2 \| \mathcal{N}^{1/2} \mathcal{K}^{1/2} \xi_\nu \|^2 + C N^{\kappa -1 } \| \gamma_{S \cup H} \|_\infty^2 \| \gamma_S \|^2 \| \mathcal{N} \xi \|^2 \end{align*} $$

with $\mathcal {V}_N^{(H)}$ defined as in (3.4). With Lemma 2.4 and Proposition 2.2, we find (using the assumption $3\kappa - 2 + 4 \varepsilon < 0$ )

$$ \begin{align*} \langle \xi_\nu, \text{G}_1 \xi_\nu \rangle \leq \langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle + C N^{5\kappa/2} \cdot N^{9 \kappa - 5 + 6\varepsilon} \| \xi_\nu \|^2. \end{align*} $$

With (4.4) and (4.5), we have shown that

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N^{(4)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \frac{\langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle}{\| \xi_\nu \|^2} + C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6 \varepsilon} \}. \end{align*} $$

Combining the last bound with (4.2) and (4.3), we obtain

$$ \begin{align*} \frac{\langle \xi_\nu, \mathcal{G}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; &\widetilde{C}_N + \frac{\langle \xi_\nu, (\mathcal{K} +\mathcal{V}_N^{(H)} + \mathcal{C}_N) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &+ C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{9\kappa - 5 + 6\varepsilon} , N^{21 \kappa /4 - 3 + 3 \varepsilon } \} \end{align*} $$

where we defined

(4.6) $$ \begin{align} \widetilde{C}_{N} =\; & \frac{N_0^2 }{2N} N^\kappa \widehat{V}(0) +\sum_{p\in {\Lambda}^*_+}p^2\sigma_p^2 + \frac{N_0}{N}\sum_{p\in \Lambda^*_+} N^\kappa \big( \widehat{V}(p/N^{1-\kappa}) + \widehat{V}(0) \big)\sigma_p^2\nonumber\\ & + \frac{N_0}{N}\sum_{p\in \Lambda^*_+}N^\kappa \widehat{V}(p/N^{1-\kappa})\sigma_p\gamma_p +\frac{1}{2N}\sum_{\substack{p,r\in \Lambda^*_+\\r\neq p}}N^\kappa \widehat{V}(r/N^{1-\kappa})\sigma_p\sigma_{p+r}\gamma_p\gamma_{p+r}. \end{align} $$

Inserting $N_0 = N - \| \sigma _L \|^2$ and recalling from Lemma 2.4 that $\| \sigma _L \|^2 \leq C N^{3\kappa /2}$ and $\| \sigma _{L^c} \|^2 \leq C N^{3\kappa /2 - \varepsilon }$ , we obtain $\widetilde {C}_N = C_{\mathcal {G}_N} + \mathcal {O} (N^{5\kappa /2 - \varepsilon })$ , with $C_{\mathcal {G}_N}$ as defined in (3.6) (with the assumption $3\kappa - 2 + 4\varepsilon < 0$ ). To handle the first term on the second line of (4.6), we used that $|\sigma _p \gamma _p - \eta _p| \leq C \eta _p^3 \leq C N^{3\kappa }/|p|^6$ , for $p \in P_L^c$ . This completes the proof of Proposition 3.1.

5 Cubic conjugation

In this section we prove Proposition 2.2 and Proposition 3.2, as a consequence of the following lemma.

Lemma 5.1. Let $A_\nu $ be defined in (2.15) and $\mathcal {K}$ , $\mathcal {V}_N^{(H)}$ and $\mathcal {C}_N$ be defined in (3.4) and (3.5), respectively. Then, for $\xi _\nu =e^{A_\nu }\Omega $ ,

(5.1) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq\; & \frac 2 {N} \sum_{\substack{v \in P_S, r \in P_H:\\r+v \in P_H}} r^2\eta_r (\eta_r+\eta_{r+v})\sigma_v^2 + \mathcal{E} , \end{align} $$

(5.2) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{C}_N \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \; & \frac 2 {N} \sum_{\substack{v \in P_S, r \in P_H :\\ r+v \in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r +\eta_{r+v}) \sigma_v^2 + \mathcal{E} , \end{align} $$

(5.3) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{V}_N^{(H)} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \;& \frac 1 {N^2} \sum_{\substack{v \in P_S, r \in P_H: \\ r+v \in P_H}} \, \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_r (\eta_r+\eta_{r+v}) \sigma_v^2 + \mathcal{E} , \end{align} $$

with

$$ \begin{align*} \mathcal{E} \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon},\, N^{12\kappa-7 +5\varepsilon} \} \end{align*} $$

for all $\kappa \in (1/2 ;2/3)$ , $\varepsilon>0$ so small that $3\kappa - 2 + 4 \varepsilon < 0$ and N large enough.

With Lemma 5.1, we can immediately show Proposition 3.2.

Proof Proof of Proposition 3.2.

From Lemma 5.1 we have

$$ \begin{align*} &\frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ & \leq \frac 2 N \sum_{v \in P_S} \sigma_v^2 \sum_{\substack{r \in P_H: \\ r+v \in P_H}}\Big[ r^2 \eta_r + N^\kappa \widehat V(r/N^{1-\kappa}) + \frac {N^\kappa} {2N} \big(\widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_r \Big] (\eta_r+\eta_{r+v}) + \mathcal{E} \end{align*} $$

with $\mathcal {E} \leq CN^{5\kappa /2} \cdot \max \{ N^{-\varepsilon }, N^{12\kappa -7 + 5\varepsilon } \}$ . With the scattering equation (2.9), we obtain

$$ \begin{align*} \frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq \frac1N\sum_{v\in P_S}\sigma_v^2\sum_{\substack{r \in P_H: \\r+v\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v})+ \mathcal{E}' \end{align*} $$

with

$$ \begin{align*} \mathcal{E}' \leq \; \frac 1 N \sum_{v \in P_S} \sigma_v^2 \sum_{\substack{r \in P_H:\, r+v \in P_H}} N^{3-2\kappa} \lambda_{\ell} (\widehat \chi_\ell \ast \widehat f_{N,\ell})_r \eta_r + \mathcal{E} . \end{align*} $$

Using $|N^{3-3\kappa }\lambda _{\ell }|\leq C$ and $\|\widehat {\chi }_\ell \ast \widehat f_{N,\ell }\|\leq C$ , we conclude

$$ \begin{align*} &\frac{\langle \xi_\nu, (\mathcal{K} + \mathcal{V}_N^{(H)} + \mathcal{C}_{N} ) \xi_\nu \rangle}{\| \xi_\nu \|^2} \\ &\hspace{.3cm} \leq \frac1N\sum_{v\in P_S}\sigma_v^2\sum_{\substack{r \in P_H: \\r+v\in P_H}} N^\kappa \widehat V(r/N^{1-\kappa})(\eta_r+\eta_{r+v}) + C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon}, N^{12\kappa -7 + 5\varepsilon} \}. \end{align*} $$

Finally, with (3.12) and the expression (2.28) for $\sigma ^2_v$ , we can extend the sum over $v \in P_S$ to a sum over all $v \in P_L$ , without changing the size of the error. This completes the proof of Proposition 3.2.

We still have to show Proposition 2.2 and Lemma 5.1.

5.1 Expectation of the particle number and kinetic energy

In this section we prove (5.1) and Proposition 2.2. We start by computing the expectation $\langle \xi _\nu , \mathcal {K} \xi _\nu \rangle $ . We proceed as we did in (2.19)–(2.22) to compute $\| \xi _\nu \|^2$ . With $\mathcal {K} a_{r+v}^* a_{-r}^* a_{-v}^* = a_{r+v}^* a_{-r}^* a_{-v}^* (\mathcal {K} + (r+v)^2 + r^2 + v^2)$ we obtain

$$ \begin{align*} {\bigl\langle\xi_{\nu},\mathcal{K} \xi_{\nu}\bigr\rangle} = &\; \sum_{m\geq 1}\frac1{2^m (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta \big( \{ r_j, v_j \}_{j=1}^{m} \big) \\&\hspace{3cm} \times [r_m^2+v_m^2+(r_m+v_m)^2]\, \prod_{i=1}^m (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \, \end{align*} $$

with the cutoff $\theta $ introduced in (2.18). Since all terms are positive, we can find an upper bound for $\langle \xi _\nu , \mathcal {K} \xi _\nu \rangle $ by replacing $\theta (\{r_j,v_j\}_{j=1}^m)$ with $\theta (\{r_j,v_j\}_{j=1}^{m-1})$ , removing conditions involving momenta with index m. Recalling (2.22), we find

$$ \begin{align*} {\bigl\langle\xi_{\nu},\mathcal{K} \xi_{\nu}\bigr\rangle} &\leq \frac{1}{2N} \sum_{\substack{v\in P_S,r\in P_H:\\r+v\in P_H}}[r^2+v^2+(r+v)^2] (\eta_r + \eta_{r+v})^2 \sigma_v^2 \,\|\xi_\nu\|^2 \\ &\leq \frac{2}{N} \sum_{v \in P_S , r\in P_H} r^2 \eta_r(\eta_r + \eta_{r+v}) \sigma_v^2 \,\|\xi_\nu\|^2 + \mathcal{E} \end{align*} $$

with (using Lemma 2.4 and the assumption $3\kappa -2 +4\varepsilon < 0$ )

$$ \begin{align*} \frac{\mathcal{E}}{\|\xi_\nu\|^2} =& \frac2N \sum_{\substack{v\in P_S,r\in P_H : \\ r+v \in P_H}} \hspace{-.3cm} (v ^2+r\cdot v) \eta_r (\eta_r+\eta_{r+v})\sigma_v^2 \\&\leq \frac C N( \| \sigma_S \|_{H^1}^2 \| \eta_H \|^2 +\|\sigma_S\|\|\eta_H\|\|\sigma_S\|_{H^1}\|\eta_H\|_{H^1})\leq C N^{4\kappa -1+\varepsilon}\leq C N^{5\kappa/2-\varepsilon}. \end{align*} $$

This proves (5.1). In particular, (5.1) implies, together with Lemma 2.4, that

(5.4) $$ \begin{align} \frac{\langle \xi_\nu , \mathcal{K} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq C N^{-1} \| \eta_H \|_{H^1}^2 \| \sigma_S \|^2 \leq C N^{5\kappa /2}, \end{align} $$

which shows (2.24) with $j=1$ in Proposition 2.2.

Analogously, we find

$$ \begin{align*} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle & \leq \sum_{m\geq 1}\frac{3m}{2^m (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta \big( \{ r_j, v_j \}_{j=1}^{m} \big) \\&\hspace{3cm} \times [r_m^2+v_m^2+(r_m+v_m)^2]\, \prod_{i=1}^m (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 . \end{align*} $$

Writing $m = 1+ (m-1)$ and bounding $\theta ( \{ r_j, v_j \}_{j=1}^{m} )$ by $\theta ( \{ r_j, v_j \}_{j=1}^{m-2} )$ , we obtain

$$ \begin{align*} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle \leq \; & 3 \langle \xi_\nu , \mathcal{K} \xi_\nu \rangle + \frac{3}{4N^2} \sum_{r,r' \in P_H, v,v' \in P_S} [ r^2 + v^2 + (r+v)^2] \\ &\hspace{5cm} \times (\eta_r + \eta_{r+v})^2 (\eta_{r'} + \eta_{r' + v'})^2 \sigma_v^2 \sigma_{v'}^2 \| \xi_\nu \|^2. \end{align*} $$

With (5.4) and with the bounds for $\| \eta _H \|_{H^1}^2, \| \eta _H \|^2, \| \sigma _S \|^2$ from Lemma 2.4, we find

(5.5) $$ \begin{align} \frac{\langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle}{\| \xi_\nu \|^2} \leq C N^{5\kappa /2} \cdot N^{9\kappa/2 - 2 + \varepsilon}, \end{align} $$

which shows (2.24) with $j=2$ .

To show (2.23) we observe that, by (2.15), the operator $A_\nu $ only creates particles with momenta in $P_S \cup P_H$ and for each particle with momentum in $P_S$ , it creates two particles with momenta in $P_H$ . Since $|p|> N^{1-\kappa -\varepsilon }$ for all $p \in P_H$ , we find, by (5.4),

$$ \begin{align*} \langle \xi_\nu, \mathcal{N} \xi_\nu \rangle &= \sum_{p \in P_S \cup P_H} \langle \xi_\nu, a^*_p a_p \xi_\nu \rangle = \frac{3}{2} \sum_{p \in P_H} \langle \xi_\nu , a^*_p a_p \xi_\nu \rangle \\ &\leq C N^{-2+2\kappa+2\varepsilon} \langle \xi_\nu, \mathcal{K} \xi_\nu \rangle \leq N^{9\kappa/2 - 2 + 2\varepsilon} \| \xi_\nu \|^2, \end{align*} $$

proving (2.23) for $j=1$ . Analogously, we find

$$ \begin{align*} \langle \xi_\nu, \mathcal{N}^2 \xi_\nu \rangle &= \; \sum_{p\in P_S\cup P_H} \langle \mathcal{N}^{1/2}\xi_\nu, a^*_p a_p \mathcal{N}^{1/2}\xi_\nu \rangle \\ &\leq \; \frac32 \sum_{p\in P_H} \langle \mathcal{N}^{1/2}\xi_\nu, a^*_p a_p \mathcal{N}^{1/2} \xi_\nu \rangle \leq C N^{-2+2\kappa+2\varepsilon} \langle \xi_\nu, \mathcal{K} \mathcal{N} \xi_\nu \rangle . \end{align*} $$

By (5.5), we obtain (2.23) with $j=2$ . This completes the proof of Proposition 2.2.

5.2 Expectation of the cubic term

The goal of this section is to show (5.2). From (3.5), we have (using the reality of $\eta _p, \gamma _p, \sigma _p$ )

$$ \begin{align*} &\langle\xi_\nu,\mathcal{C}_N\xi_\nu\rangle\\ &=2\frac{\sqrt{N_0}}{ N}\sum_{m\geq 1}\frac1{m!(m-1)!}\sum_{\substack{p,r \in P_H\\ p+r \in P_S}} N^\kappa \widehat V(r/N^{1-\kappa})\, \sigma_{p+r} \gamma_p\gamma_r \langle A_\nu^{m} \xi_\nu, \,a^*_{p+r} a^*_{-p} a^*_{-r} A_\nu^{m-1}\xi_\nu\rangle. \end{align*} $$

Proceeding as in the previous section, we get

$$ \begin{align*}\begin{aligned} & {\bigl\langle\xi_{\nu},\mathcal{C}_N\xi_{\nu}\bigr\rangle}\\ &= 2 \sqrt{\frac{N_0}{N}}\,\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m}\sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \hskip -0.5cm \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \\ & \hskip 0.5cm \times N^\kappa \widehat V(r_m/N^{1-\kappa})\,\big(\eta_{r_m} +\eta_{r_m+v_m}\big)\gamma_{r_m}\gamma_{r_m+v_m} \sigma^2_{v_m}\, \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2. \end{aligned}\\[-17pt]\end{align*} $$

To reconstruct the norm $\| \xi _\nu \|^2$ on the right-hand side, we need to free the momenta with index m. To this end, we recall the defintion (2.18) to write

(5.6) $$ \begin{align} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) = \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big)\, \theta_m\big( \{ r_j, v_j \}_{j=1}^{m} \big)\\[-17pt]\nonumber \end{align} $$

with

$$ \begin{align*} \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big)= \prod_{i,j=1}^{m-1}\prod_{\substack{p_i,p_j,p_m:\\p_\ell\in\{-r_\ell,r_\ell+v_\ell\}}}\delta_{p_i\neq -p_j+v_m}\delta_{-p_m+v_i\neq p_j} \\[-17pt]\end{align*} $$

collecting all conditions involving $\{r_m, v_m\}$ . Writing $\theta _m = 1 + [ \theta _m - 1]$ , we split $\langle \xi _\nu , \mathcal {C}_N \xi _\nu \rangle = I_{\mathcal {C}} + J_{\mathcal {C}}$ with (recall the expression (2.22) for $\| \xi _\nu \|^2$ )

$$ \begin{align*} I_{\mathcal{C}} = \; 2\,\sqrt{\frac{N_0}{N}}\,\sum_{\substack{ v \in P_S,\, r \in P_H: \\r+v \in P_H}} N^{\kappa-1}\widehat V(r/N^{1-\kappa})\,\big(\eta_{r}+\eta_{r+v}\big)\gamma_{r}\gamma_{r+v} \sigma^2_{v}\, \| \xi_\nu\|^2 \\[-17pt]\end{align*} $$

and

$$ \begin{align*}\begin{aligned} J_{\mathcal{C}} =\; & 2\sqrt{\frac{N_0}{N}}\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m} \sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \\ & \times \Big[ \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big) -1 \Big] N^\kappa\widehat V(r_m/N^{1-\kappa})\,\big(\eta_{r_m} +\eta_{r_m+v_m}\big)\gamma_{r_m}\gamma_{r_m+v_m} \sigma^2_{v_m} \\ &\hspace{8cm} \times \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 . \end{aligned}\end{align*} $$

With $|\sqrt {N_0/N} - 1| \leq C \| \sigma _L \|^2/N$ and $|\gamma _r \gamma _{r+v} - 1| \leq C N^{2\kappa } / |r|^4$ for all $r \in P_H, v\in P_S$ , we obtain (using (3.12) and the assumption $3\kappa - 2 + 4\varepsilon < 0$ ) that

(5.7) $$ \begin{align} \frac{I_{\mathcal{C}}}{\| \xi_\nu \|^2} \leq \frac{2}{N} \sum_{\substack{ v \in P_S,\, r \in P_H: \\r+v \in P_H}} N^{\kappa}\widehat V(r/N^{1-\kappa})\,\big(\eta_{r}+\eta_{r+v}\big) \sigma^2_{v} + C N^{5\kappa /2 - \varepsilon}.\\[-17pt]\nonumber \end{align} $$

To complete the proof of (5.2), we focus now on the error term $J_{\mathcal {C}}$ . We observe that

(5.8) $$ \begin{align} | \theta_m\big( \{r_j, v_j \}_{j=1}^{m} \big) -1 | \leq& \sum_{j=1}^{m-1} \Big[ \delta_{v_j , v_m} + \sum_{ \substack{ p_m \in \{ -r_m, r_m+v_m \} \\p_j \in \{ -r_j, r_j+v_j\} }}\delta_{p_m, p_j} \Big] \nonumber\\ & + \sum_{\substack{j,k=1\\j\neq k}}^{m-1} \Big[ \sum_{ \substack{ p_j \in \{ -r_j, r_j+v_j \} \\ p_k \in \{ -r_k, r_k+v_k\} }} \delta_{v_m, p_j+p_k} + \sum_{\substack{ p_m \in \{ -r_m, r_m+v_m \} \\ p_j \in \{ -r_j, r_j+v_j \} }}\delta_{p_m, -p_j+v_k} \Big] . \end{align} $$

We bound $|J_{\mathcal {C}}| \leq \text {X}_1+ \text {X}_2$ , with $\text {X}_1$ denoting the contribution arising from the first term on the right-hand side of (5.8) (this term involves two indices, m and j) and $\text {X}_2$ indicating the contribution from the second term on the right-hand side of (5.8) (this term involves three indices, $m,j,k$ ). We can estimate

$$ \begin{align*} \text{X}_1 \leq \; &C \sum_{m\geq 2}\frac1{2^{m-2} (m-2)!}\frac1{N^m}\sum_{\substack{ v_1 \in P_S, r_1 \in P_H : \\ r_1 +v_1 \in P_H } } \cdots \sum_{\substack{ v_m \in P_S, r_m \in P_H : \\ r_m +v_m \in P_H } } \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \\ &\times N^\kappa |\widehat V(r_m/N^{1-\kappa})| \,\big| \eta_{r_m} +\eta_{r_m+v_m}\big| |\gamma_{r_m}| |\gamma_{r_m+v_m}| \sigma^2_{v_m} \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \\ & \times \Big[ \delta_{v_m, v_{m-1}} +\sum_{\substack{p_{m-1},p_m:\\p_\ell\in \{-r_\ell,r_\ell+v_\ell\}}}\hspace{-0.3cm}\delta_{p_m, p_{m-1}}\Big] . \end{align*} $$

With $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ , we reconstruct $\| \xi _\nu \|^2$ . Since $\| \gamma _H \|_\infty \leq C$ , we end up with

$$ \begin{align*} \frac{\text{X}_1}{\| \xi_\nu \|^2} \leq\; & \frac{C}{N^2} \sum_{r , r' \in P_H, v,v' \in P_S} N^\kappa |\widehat{V} (r/N^{1-\kappa}) |\eta_r + \eta_{r+v}| |\eta_{r'} + \eta_{r'+v'}|^2 \sigma_v^2 \sigma_{v'}^2 \\ &\hspace{6cm} \times \Big[ \delta_{v,v'} + \sum_{\substack{p \in \{-r, r+v\} \\ p' \in \{ -r', r' +v' \}}} \delta_{p,p'} \Big] \\ \leq \; &C N^{2\kappa-2} \| \sigma_S \|_\infty^2 \| \sigma_S \|^2 \| \eta_H \|^2 \sum_{r \in P_H} \frac{|\widehat{V} (r/N^{1-\kappa})|}{r^2} + C N^{4\kappa -2} \| \sigma_S \|^4 \sum_{r \in P_H} |r|^{-6} \\ \leq \; &C N^{11\kappa/2-2+2\varepsilon} + C N^{10\kappa-5+3\varepsilon} \leq C N^{5\kappa/2-\varepsilon}, \end{align*} $$

where we used Lemma 2.4, (3.12), the assumption $3\kappa -2 + 4\varepsilon < 0$ and the remark that $|\eta _{r+v}| \leq CN^\kappa |r|^{-2}$ , for all $r \in P_H$ and $v \in P_S$ . We can proceed similarly to estimate $X_2$ . In the second term on the right-hand side of (5.8), we have to sum over $(m-1) (m-2)/2$ pairs of indices $j,k$ . With $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-3} \big )$ and again with Lemma 2.4 and(3.12), we arrive at

$$ \begin{align*} \frac{\text{X}_2}{\| \xi_\nu \|^2} \leq\; & \frac{C}{N^3} \sum_{\substack{r , r' , r'' \in P_H, \\ v,v', v'' \in P_S}} N^\kappa |\widehat{V} (r/N^{1-\kappa}) |\eta_r + \eta_{r+v}| |\eta_{r'} + \eta_{r'+v'}|^2 |\eta_{r''} + \eta_{r''+v''}|^2 \sigma_v^2 \sigma_{v'}^2 \sigma_{v''}^2 \\ &\hspace{4.5cm} \times \Big[ \sum_{ \substack{p \in \{ -r , r+v \}, \\ p' \in \{ -r', r' + v'\}}} \delta_{p,-p'+v''} + \sum_{\substack{p' \in \{-r', r'+v'\}, \\ p'' \in \{-r'', r'' + v''\}}} \delta_{v,p'+p''} \Big] \\ \leq \; &C N^{4\kappa - 3} \| \sigma_S \|^6 \| \eta_H \|^2 \sum_{r \in P_H} |r|^{-6} + C N^{6\kappa -3} \| \sigma_S \|^6 \sum_{r \in P_H} \frac{|\widehat{V} (r/N^{1-\kappa})|}{r^2} \sum_{r' \in P_H} |r'|^{-8} \\ \leq \; &C N^{29\kappa/2 -7 + 5\varepsilon} \leq C N^{5\kappa/2} \cdot N^{12 \kappa -7 + 5\varepsilon} . \end{align*} $$

Thus, $|J_{\mathcal {C}}| / \| \xi _\nu \|^2 \leq N^{5\kappa /2} \cdot \max \{ N^{-\varepsilon } , N^{12\kappa -7 + 5\varepsilon } \}$ . With (5.7), this implies (5.2).

5.3 Expectation of the quartic term

In this section we show the bound (5.3) for the expectation of $\mathcal {V}_N^{(H)}$ . Pairing momenta in $P_S$ , as we did in (2.20) and in the previous subsections, we obtain

(5.9) $$ \begin{align} \begin{aligned} {\bigl\langle\xi_\nu,\mathcal{V}_N^{(H)} \xi_\nu\bigr\rangle}=&\frac1{2N}\sum_{m\geq 1}\frac1{m!}\frac1{N^m} \sum_{\substack{v_1 \in P_S,\, r_1, {\tilde{r}}_1 \in P_H:\\ r_1+v_1,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots\sum_{\substack{v_m \in P_S,\, r_m, {\tilde{r}}_m \in P_H:\\ r_m+v_m,\, {\tilde{r}}_m + v_m \in P_H}} \\ &\times \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \prod_{i=1}^m\eta_{r_i}\eta_{\tilde{r}_i}\sigma_{v_i}^2\hspace{-0.3cm}\sum_{\substack{r\in \Lambda^*, p,q \in P_H: \\p+r,\,q+r\in P_H}}\hspace{-0.2cm} N^\kappa \widehat V(r/N^{1-\kappa})\\ &\times{\bigl\langle\Omega,A_{r_1,v_1}\dots A_{r_m,v_m}\,a^*_{p+r}a^*_{q} a_p a_{q+r} A^*_{\tilde{r}_1,v_1}\dots A^*_{\tilde{r}_m,v_m}\Omega\bigr\rangle} \end{aligned}\end{align} $$

where we use the notation $A_{r_i, v_i} = a_{r_i+v_i} a_{-r_i}$ that was already introduced in (2.20). Next we observe that because of the cutoffs $\theta (\{r_j,v_j\}_{j=1}^m)$ and $\theta (\{{\tilde {r}}_j,v_j\}_{j=1}^m)$ , at most two indices $i,j \in \{ 1, \dots , m \}$ can be involved in contractions with the observable $a_{p+r}^* a_q^* a_p a_{q+r}$ . We distinguish two possible cases:

1. 1) There exists an index $i \in \{ 1,\dots ,m \}$ such that $a_p$ , $a_{q+r}$ are contracted with $A^*_{{\tilde {r}}_i,v_i}$ and $a^*_{q}$ , $a^*_{p+r}$ are contracted with $A_{r_i,v_i}$ .

2. 2) There are two indices $i \neq j \in \{ 1, \dots , m \}$ such that the operators $a_p$ and $a_{q+r}$ are contracted with $a^*_{{\tilde {p}}_i}$ and $a^*_{{\tilde {p}}_j}$ for some ${\tilde {p}}_\ell \in \{-{\tilde {r}}_\ell , {\tilde {r}}_\ell + v_\ell \}, \ell =i,j$ and the operators $a^*_q, a^*_{p+r}$ are contracted with $a_{p_i}, a_{p_j}$ , with $p_\ell \in \{-r_\ell ,r_\ell +v_\ell \}, \ell =i,j$ . Note that in this case the operators $a^*_{-{\tilde {p}}_i+v_i}, a^*_{-{\tilde {p}}_j+v_j}$ have to be contracted with $a_{-p_i+v_i}, a_{-p_j+v_j}.$

We denote by $\text {V}_1$ and $\text {V}_2$ the contributions to ${\bigl \langle \xi _\nu ,\mathcal {V}_N^{(H)} \xi _\nu \bigr \rangle }$ arising from the two cases described above. Let us first consider $\text {V}_1$ . There are m choices (all leading to the same contribution) for the index $i \in \{ 1, \dots , m \}$ labelling momenta to be contracted with the observable. Let us fix $i = m$ . Then we have $p={\tilde {p}}_m, q+r=-{\tilde {p}}_m+v_m$ with ${\tilde {p}}_m \in \{-{\tilde {r}}_m,{\tilde {r}}_m +v_m \}$ and $p+r=p_m, q=-p_m+v_m$ with $p_m\in \{-r_m,r_m+v_m\}$ . Note that the choice of p and $p+r$ also determines q and $q+r$ , since we always have $q = v_m - (p+r)$ . The presence of the cutoffs immediately implies that $A_{r_j,v_j}$ is fully contracted with $A^*_{{\tilde {r}}_j , v_j}$ , for all $j \not = m$ . We find

(5.10) $$ \begin{align} &\langle \xi_{\nu}, \text{V}_1 \xi_{\nu} \rangle \nonumber\\ &=\frac1{2N}\sum_{m\geq 1}\frac1{(m-1)!}\frac1{N^m} \;\sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \nonumber\\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m} \in P_S, r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}}\hskip -0.5cm\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \nonumber\\[6pt] &\, \times \hskip -0.1cm \prod_{j=1}^{m-1}\hskip -0.1cm\eta_{r_j}\eta_{{\tilde{r}}_j}(\delta_{r_j,{\tilde{r}}_j}+\delta_{-r_j,{\tilde{r}}_j+v_j})\sigma_{v_j}^2\, \eta_{r_m} \eta_{{\tilde{r}}_m}\sigma_{v_m}^2\hspace{-.5cm} \sum_{\substack{r\in \Lambda^*,\,p\in P_H:\\p-v_m,p+r\in P_H}} \hskip -0.7cm N^\kappa\widehat{V}(r/N^{1-\kappa}) \hskip -0.5cm \sum_{\substack{p_m\in \{-r_m,r_m+v_m\}\\{\tilde{p}}_m\in \{-{\tilde{r}}_m,{\tilde{r}}_m+v_m\}}}\hspace{-0.8cm}\delta_{p,p_m}\delta_{p+r,{\tilde{p}}_m} . \end{align} $$

Since here (in contrast to the previous subsections) the contraction does not fix ${\tilde {r}}_m$ to be either $r_m$ or $-(r_m + v_m)$ , we cannot erase the cutoff $\theta (\{ {\tilde {r}}_j , v_j \}_{j=1}^m)$ . With the decomposition (5.6), we can replace, on the right-hand side of (5.10),

$$ \begin{align*} \theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) = \theta \big( \{r_j, v_j \}_{j=1}^{m-1} \big) \theta_m \big( \{ r_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big). \end{align*} $$

Writing

$$ \begin{align*} \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) = 1 + \Big[ \theta_m \big( \{ r_j, v_j \}_{j=1}^{m} \big) \theta_m \big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) - 1 \Big] \end{align*} $$

we split (as we did in the last subsection) $\langle \xi _\nu , \text {V}_1 \xi _\nu \rangle = I_{\mathcal {V}} + J_{\mathcal {V}}$ , with

(5.11) $$ \begin{align} I_{\mathcal{V}} = \frac 1 {N^2} \sum_{r \in \Lambda^*} \sum_{\substack{ v\in P_S, p \in P_H:\\ p+r,p-v,\, p+r-v \in P_H}} \hskip -0.5cm N^\kappa \widehat V(r/N^{1-\kappa}) \eta_{p} \big( \eta_{p+r} +\eta_{p+r-v} \big) \sigma_{v}^2 \, \| \xi_\nu \|^2 \end{align} $$

and

(5.12) $$ \begin{align} J_{\mathcal{V}} =&\frac1{2N}\sum_{m\geq 1}\frac1{2^{m-1} (m-1)!}\frac1{N^m}\sum_{\substack{v_1 \in P_S, r_1 \in P_H: \\ r_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m-1} \in P_S, r_{m-1} \in P_H: \\ r_{m-1} + v_{m-1} \in P_H}} \sum_{\substack{v_{m} \in P_S , r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}} \nonumber\\ & \times \theta\big( \{r_j, v_j \}_{j=1}^{m-1} \big) \big[ \theta_m(\{r_j, v_j\}_{j=1}^{m})\theta_m(\{ r^\sharp_j ,v_j\}_{j=1}^m)-1\big] \prod_{i=1}^{m-1} (\eta_{r_i} + \eta_{r_i + v_i})^2 \sigma_{v_i}^2 \, \nonumber\\ &\times \eta_{r_m} \eta_{{\tilde{r}}_m} \sigma_{v_m}^2 \sum_{\substack{r\in \Lambda^*}} N^\kappa \widehat V(r/N^{1-\kappa})\hspace{-0.5cm} \sum_{\substack{ p \in P_H :\, p+r,\\ p-v_m,\, p+r-v_m\in P_H}} \sum_{\substack{p_m\in \{-r_m,r_m+v_m\}\\{\tilde{p}}_m\in \{-{\tilde{r}}_m,{\tilde{r}}_m+v_m\}}}\hspace{-0.3cm}\delta_{p,p_m}\delta_{p+r,{\tilde{p}}_m} \end{align} $$

where $r^\sharp _j = r_j$ for $j=1,\dots , m-1$ and $r^\sharp _m = {\tilde {r}}_m$ in the argument of $\theta _m$ . Observing that, with Lemma 2.4 and (3.12),

$$ \begin{align*} \frac 1 {N^2} \sum_{r \in \Lambda^*} & \sum_{\substack{v \in P_S, p \in \Lambda^* : \\ p\in P_H^c \text{ or } p-v \in P_H^c}} N^\kappa |\widehat V(r/N^{1-\kappa})| |\eta_{p}| \big( |\eta_{p+r}| + |\eta_{p+r-v}| \big) \sigma_{v}^2 \\ &\leq C N^{-2+2\kappa} \| \sigma_S \|^2 \Big[ \sum_{|p| \leq N^{1-\kappa -\varepsilon}} |p|^{-2} \Big] \mathop{\mathrm{sup}}\limits_{p\in \Lambda^*} \sum_{r \in \Lambda^*} |\widehat V(r/N^{1-\kappa})| |\eta_{p+r}| \leq C N^{5\kappa/2 - \varepsilon}, \end{align*} $$

we conclude from (5.11) (switching $p+r \to p$ and $v \to -v$ ) that

(5.13) $$ \begin{align} \frac{I_{\mathcal{V}}}{\| \xi_\nu \|^2} \leq \frac 1 {N^2} \sum_{\substack{v \in P_S, p \in P_H: \\ p+v \in P_H}} \, \big( N^\kappa \widehat V(\cdot/N^{1-\kappa}) \ast \eta \big)_p (\eta_p +\eta_{p+v}) \sigma_v^2 + C N^{5\kappa/2-\varepsilon}. \end{align} $$

Let us now focus on the term $J_{\mathcal {V}}$ . With

(5.14) $$ \begin{align} \Big| \theta_m \big( &\{r_j, v_j \}_{j=1}^{m} \big) \theta_m\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) -1 \Big| \nonumber\\ \leq \; &\sum_{j=1}^{m-1} \delta_{v_m, v_j} + \sum_{j=1}^{m-1} \Big[ \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_m \in \{ - r_m , r_m + v_m \}}} \delta_{p_m, p_j} + \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ {\tilde{p}}_m \in \{ - {\tilde{r}}_m , {\tilde{r}}_m + v_m \}}} \delta_{{\tilde{p}}_m, p_j} \Big] \nonumber \\ &+ \sum_{\substack{j,k=1 \\ j \not = k}}^{m-1} \Big[ \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_m \in \{ - r_m , r_m + v_m \}}} \delta_{p_m, -p_j + v_k} + \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ {\tilde{p}}_m \in \{ - {\tilde{r}}_m , {\tilde{r}}_m + v_m \}}} \delta_{{\tilde{p}}_m, -p_j + v_k} \Big]\nonumber \\ &+ \sum_{\substack{j,k=1 \\ j \not = k}}^{m-1} \sum_{\substack{p_j \in \{ - r_j, r_j + v_j \} \\ p_k \in \{ - r_k , r_k + v_k \}}} \delta_{v_m , p_j + p_k} \end{align} $$

we can bound $|J_{\mathcal {V}}| \leq \text {W}_1 + \text {W}_2 + \text {W}_3 + \text {W}_4$ , with $W_\ell $ indicating the contribution to (5.12) arising from the $\ell $ th term on the right-hand side of (5.14).

The term $\text {W}_1$ contains the sum of $(m-1)$ identical contributions, corresponding to $j \in \{ 1, \dots , m-1 \}$ in the first term on the right-hand side of (5.14). Let us fix $j=m-1$ . Estimating $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ and reconstructing the expression (2.22) for $\| \xi _\nu \|^2$ , we can bound (the momenta $r',r'',\tilde {r}''$ correspond to $r_{m-1}, r_m, \tilde {r}_m$ )

$$ \begin{align*} \frac{\text{W}_1}{\| \xi_\nu \|^2} \leq C N^{-3} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \hspace{-.3cm} \sum_{\substack{r', r'', {\tilde{r}}'' \in P_H \\ v' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^4 \hspace{-.4cm} \sum_{\substack{p'' \in \{ -r'', r''+v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v' \}}} \hspace{-.5cm} \delta_{p''+r ,{\tilde{p}}''}. \end{align*} $$

With Lemma 2.4 and with the estimate

(5.15) $$ \begin{align} \mathop{\mathrm{sup}}\limits_{v \in P_S \cup \{ 0 \}} \frac{1}{N^2} \sum_{\substack{r \in \Lambda^*, q \in P_H : \\ q-r \in P_H}} N^\kappa |\widehat{V} (r/N^{1-\kappa})| |\eta_{q-v} | |\eta_{q-r}| \leq C N^{\kappa}, \end{align} $$

which can be shown similarly to (3.12) (using $V \in L^q (\mathbb {R}^3)$ , for some $q> 3/2$ ), we find

(5.16) $$ \begin{align} \frac{\text{W}_1}{\| \xi_\nu \|^2} \leq C N^{\kappa -1} \| \eta_H \|^2 \| \sigma_S \|_\infty^2 \| \sigma_S \|^2 \leq C N^{11\kappa/2 -2 + 2\varepsilon} \leq C N^{5\kappa/2-\varepsilon} \end{align} $$

since $3\kappa - 2 + 4\varepsilon < 0$ . Analogously, we bound, with (3.12) and Lemma 2.4,

(5.17) $$ \begin{align} \frac{\text{W}_2}{\| \xi_\nu \|^2} \leq \; &C N^{-3} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \hspace{-.3cm} \sum_{\substack{r', r'', {\tilde{r}}'' \in P_H \\ v', v'' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^2 \sigma_{v''}^2 \hspace{-.3cm}\nonumber\\ &\times \sum_{\substack{p'' \in \{ -r'', r''+v'' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v'' \}}} \hspace{-.3cm} \delta_{p'' + r , {\tilde{p}}''} \Big[ \sum_{\substack{p' \in \{ -r', r'+v' \} \\ p'' \in \{ -r'', r'' +v'' \}}} \delta_{p', p''} + \sum_{\substack{p' \in \{ -r', r'+v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'', {\tilde{r}}'' +v'' \}}} \delta_{p', {\tilde{p}}''} \Big] \nonumber\\ \leq \; &C N^{-3+5\kappa} \| \sigma_S \|^4 \Big[\sum_{r' \in P_H} |r'|^{-6}\Big] \Big[ \mathop{\mathrm{sup}}\limits_{r' \in \Lambda^*} \sum_{r \in \Lambda^* , r \not = -r'} \frac{|\widehat{V} (r/N^{1-\kappa})|}{|r+r'|^{2}} \Big] \leq C N^{5\kappa/2-\varepsilon}. \end{align} $$

As for $\text {W}_3$ , there are $(m-1)(m-2)$ possible choices of the indices $j,k$ in (5.14), all leading to the same contribution. We fix $j = m-1$ and $k =m-2$ . Estimating now $\theta \big ( \{r_j, v_j \}_{j=1}^{m-1} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-3} \big )$ , we obtain, with (3.12),

(5.18) $$ \begin{align} \frac{\text{W}_3}{\| \xi_\nu \|^2} \leq \; &C N^{-4} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \nonumber\\ &\times \sum_{\substack{r', r'', r''', {\tilde{r}}''' \in P_H \nonumber\\ v', v'', v''' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 (\eta_{r''} + \eta_{r''+v''})^2 |\eta_{r'''}| |\eta_{{\tilde{r}}'''}| \sigma_{v'}^2 \sigma_{v''}^2 \sigma_{v'''}^2 \hspace{-.5cm}\nonumber\\ &\times \sum_{\substack{p''' \in \{ -r''', r'''+ v''' \} \nonumber\\ {\tilde{p}}''' \in \{ -{\tilde{r}}''' , {\tilde{r}}''' + v''' \}}} \hspace{-.5cm} \delta_{p'''+r , {\tilde{p}}'''} \Big[ \sum_{\substack{p' \in \{ -r', r'+v' \}\nonumber \\ p''' \in \{ -r''', r''' +v''' \}}} \hspace{-.5cm} \delta_{p''', -p'+v''} + \sum_{\substack{p' \in \{ -r', r'+v' \}\nonumber \\ {\tilde{p}}''' \in \{ -{\tilde{r}}''', {\tilde{r}}''' +v''' \}}} \hspace{-.3cm} \delta_{{\tilde{p}}''' , - p' + v''} \Big] \\ \leq \; &C N^{-3+ 4\kappa} \| \sigma_S \|^6 \| \eta_H \|^2 \sum_{r' \in P_H} |r'|^{-6} \leq C N^{5\kappa/2} \cdot N^{12\kappa - 7 +4\varepsilon}. \end{align} $$

Analogously, with Lemma 2.4 and (5.15), we find

$$ \begin{align*} \frac{\text{W}_4}{\| \xi_\nu \|^2} \leq \; &C N^{-4} \sum_{r \in \Lambda^*} N^\kappa |\widehat{V} (r/N^{1-\kappa})| \\ &\times \sum_{\substack{r', r'', r''', {\tilde{r}}''' \in P_H \\ v', v'', v''' \in P_S}} \hspace{-.3cm} (\eta_{r'} + \eta_{r'+v'})^2 (\eta_{r''} + \eta_{r''+v''})^2 |\eta_{r'''}| |\eta_{{\tilde{r}}'''} | \sigma_{v'}^2 \sigma_{v''}^2 \sigma_{v'''}^2 \hspace{-.5cm}\\ &\times \sum_{\substack{p''' \in \{ -r''', r'''+ v''' \} \\ {\tilde{p}}''' \in \{ -{\tilde{r}}''' , {\tilde{r}}'''+ v''' \}}} \hspace{-.5cm} \delta_{p''' + r , {\tilde{p}}''' } \sum_{\substack{p' \in \{ -r', r'+v' \} \\ p'' \in \{ -r'', r'' +v'' \}}} \hspace{-.5cm} \delta_{v''', p' +p''} \\ \leq \; &C N^{-2+5\kappa}\| \sigma_S \|^6 \sum_{r' \in P_H} |r'|^{-8} \leq C N^{5\kappa/2} \cdot N^{12\kappa -7 + 5\varepsilon}. \end{align*} $$

Together with (5.16), (5.17), (5.18), we conclude that

(5.19) $$ \begin{align} | J_{\mathcal{V}} | \leq C N^{5\kappa/2} \cdot \max \{ N^{-\varepsilon} , N^{12\kappa - 7 + 5\varepsilon} \}. \end{align} $$

Finally, we consider the term $\text {V}_2$ , associated with the second case listed after (5.9). We fix $i =m$ and $j=m-1$ and we consider all possible contractions of $a_p$ with $a^*_{{\tilde {p}}_m}$ , of $a_{q+r}$ with $a^*_{{\tilde {p}}_{m-1}}$ and of $a_q^*, a_{p+r}^*$ with $a_{p_m}, a_{p_{m-1}}$ , where ${\tilde {p}}_\ell \in \{-{\tilde {r}}_\ell ,{\tilde {r}}_\ell +v_\ell \}$ and $p_\ell \in \{ -r_\ell , r_\ell + v_\ell \}$ , for $\ell =m,m-1$ . We obtain

$$ \begin{align*}\begin{aligned} &{\bigl\langle\xi_\nu,\text{V}_2\, \xi_\nu\bigr\rangle} \\ & = \frac{1}{2N} \sum_{m\geq 2}\frac1{(m-2)!}\frac1{N^m} \sum_{\substack{v_1 \in P_S, r_1, {\tilde{r}}_1 \in P_H: \\ r_1 + v_1 ,\, {\tilde{r}}_1 + v_1 \in P_H}} \cdots \sum_{\substack{v_{m} \in P_S, r_{m}, {\tilde{r}}_{m} \in P_H: \\ r_{m} + v_{m} ,\, {\tilde{r}}_{m} + v_{m} \in P_H}}\hskip -0.5cm\theta\big( \{r_j, v_j \}_{j=1}^{m} \big) \theta\big( \{{\tilde{r}}_j, v_j \}_{j=1}^{m} \big) \\ & \hskip 0.5cm \times \prod_{i=1}^{m-2}\eta_{r_i}\eta_{\tilde{r}_i} \big( \delta_{{\tilde{r}}_i,r_i}+\delta_{-{\tilde{r}}_i,r_i+v_i}\big)\sigma_{v_i}^2 \, \prod_{j=m,m-1} \eta_{r_j} \eta_{{\tilde{r}}_j} \sigma_{v_j}^2\\ &\hspace{0.5cm}\times \sum_{\substack{r \in \Lambda^*,\, p,q \in P_H: \\ p-r, q-r \in P_H }} N^\kappa \widehat V(r/N^{1-\kappa}) \sum_{\substack{{\tilde{p}}_\ell \in \{ -{\tilde{r}}_\ell, {\tilde{r}}_\ell+v_\ell\} \\ \ell=m-1,m } } \delta_{p, {\tilde{p}}_m} \delta_{q+r, {\tilde{p}}_{m-1}} \\ & \hspace{0.5cm} \times \hskip -0.5cm \sum_{\substack{ p_\ell \in \{ - r_\ell, r_\ell+v_\ell\} \\ \ell=m-1,m } } \big(\delta_{q, p_m} \delta_{p+r, p_{m-1}} + \delta_{q, p_{m-1}} \delta_{p+r, p_{m}} \big) \big( \delta_{{\tilde{p}}_m, p_m} + \delta_{-{\tilde{p}}_m+v_m, -p_{m-1}+v_{m-1}} \big). \end{aligned}\end{align*} $$

Estimating $\theta \big ( \{r_j, v_j \}_{j=1}^{m} \big ) \theta \big ( \{{\tilde {r}}_j, v_j \}_{j=1}^{m} \big ) \leq \theta \big ( \{r_j, v_j \}_{j=1}^{m-2} \big )$ and using Lemma 2.4 and the condition $3\kappa - 2 + 4\varepsilon < 0$ , we find

$$ \begin{align*}\begin{aligned} \frac{|{\bigl\langle\xi_\nu,\text{V}_2\, \xi_\nu\bigr\rangle}|}{\| \xi_\nu \|^2} \\ \leq \; & C N^{-3} \sum_{r \in \Lambda^*} N^\kappa \widehat{V} (r/N^{1-\kappa}) \sum_{\substack{r',{\tilde{r}}',r'', {\tilde{r}}'' \in P_H \\ v',v'' \in P_S}} |\eta_{r'}| |\eta_{{\tilde{r}}'}| |\eta_{r''}| |\eta_{{\tilde{r}}''}| \sigma_{v'}^2 \sigma_{v''}^2 \\ &\times \sum_{\substack{p' \in \{ -r' , r' + v' \} \\ p'' \in \{ -r'' , r'' + v'' \}}} \sum_{\substack{{\tilde{p}}' \in \{ -{\tilde{r}}' , {\tilde{r}}' + v' \} \\ {\tilde{p}}'' \in \{ -{\tilde{r}}'' , {\tilde{r}}'' + v'' \} }} (\delta_{{\tilde{p}}', p''+r} \delta_{{\tilde{p}}'' + r, p'} + \delta_{{\tilde{p}}', p'+r} \delta_{{\tilde{p}}'' + r, p''}) \\ &\hspace{6cm} \times (\delta_{{\tilde{p}}'', p''} + \delta_{-{\tilde{p}}'' + v'', -p'+v'}) \\ \leq \; &C N^{-3+\kappa} \| \eta_H \|^4 \| \sigma_S \|^4 \leq C N^{10\kappa - 5 + 2\varepsilon} \leq C N^{5\kappa/2 -\varepsilon}. \end{aligned} \end{align*} $$

With (5.13) and (5.19), we obtain (5.3).