The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics.

In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions.

In this article, we introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.

Let $\Omega \subset \mathbb {R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$. Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, where $\mu \in {\mathbb {R}}$. We study boundary value problems ($P_\pm$) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$, where $p>1$, $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$. Different critical exponents for the existence of a solution to ($P_\pm$) appear according to concentration of $\nu$. The solvability for problem ($P_+$) was proved in [3, 29] in subcritical ranges for $p$, namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$, we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ($P_+$) in supercritical ranges for $p$, i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ($P_-$) under a smallness assumption on $\nu$.

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}

$a, \epsilon, \eta \gt 0$

$\lambda\in \mathbb{R}$

$p=2N/(N-2)$

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation

$$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$

$\Omega =\mathbb {R}^N$

$\mathbb {R}^N\setminus \Omega$

$\rho >0$

$V\ge 0$

$V\equiv 0$

$p\in (2,2+\frac 4 N)$

$\bar u$

$V$

$\|V\|_{L^q}$

$q\ge \frac N2$

$q>1$

$N=2$

$V$

$\mathbb {R}^N\setminus \Omega$

$\bar u$

$V\equiv 0$

$\Omega =\mathbb {R}^N$

We prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form

\[ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \]

$p\ge 4$

$\gamma,b>0$

$V$

In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.

First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.

As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.

We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation

$$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux, we establish a Hardy-type inequality. In the regime with an infinite discrete spectrum, we obtain sharp spectral asymptotics with a refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

Our first aim in this paper is to deal with the maximum principle for subfunctions in an arbitrary unbounded domain. As an application, we next give a result concerning the classical Phragmén–Lindelöf theorem for subfunctions in a cone. For a subfunction defined in a cone that is dominated on the boundary by a certain function, we finally generalize the Phragmén–Lindelöf type theorem by making a generalized harmonic majorant of it.

Let $H=-\unicode[STIX]{x1D6E5}+V$ be a Schrödinger operator with some general signed potential $V$. This paper is mainly devoted to establishing the $L^{q}$-boundedness of the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ for $q>2$. We mainly prove that under certain conditions on $V$, the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,p_{0})$ with a given $2<p_{0}<n$. As an application, the main result can be applied to the operator $H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where $V_{+}$ belongs to the reverse Hölder class $B_{\unicode[STIX]{x1D703}}$ and $V_{-}\in L^{n/2,\infty }$ with a small norm. In particular, if $V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$ for some positive number $\unicode[STIX]{x1D6FE}$, $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,n/2)$ and $n>4$.

Let $A\,:=\,-\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,.\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,+\,V$ be a magnetic Schrödinger operator on ${{\mathbb{R}}^{n}}$, where

$$\vec{a}:=\left( {{a}_{1}},...,{{a}_{n}} \right)\in L_{\text{loc}}^{2}\left( {{\mathbb{R}}^{n}},{{\mathbb{R}}^{n}} \right)\operatorname{and}0\le V\in L_{\text{loc}}^{1}\left( {{\mathbb{R}}^{n}} \right)$$

satisfy some reverse Hölder conditions. Let $\phi :\,{{\mathbb{R}}^{n}}\,\times \,[0,\,\infty )\,\to \,[0,\,\infty )$ be such that $\phi \left( x,\,. \right)$ for any given $x\,\in \,{{\mathbb{R}}^{n}}$ is an Orlicz function, $\phi \left( ^{.}\,,\,t \right)\,\in \,{{\mathbb{A}}_{\infty }}\left( {{\mathbb{R}}^{n}} \right)$ for all $t\,\in \,\left( 0,\,\infty \right)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I\left( \phi \right)\,\in \,(0,\,1]$. In this article, the authors prove that second-order Riesz transforms $V{{A}^{-1}}$ and ${{\left( \nabla \,-\,i\overrightarrow{a} \right)}^{2}}{{A}^{-1}}$ are bounded from the Musielak–Orlicz–Hardy space ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$, associated with $A$, to theMusielak–Orlicz space ${{L}^{\phi }}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, we establish the boundedness of $V{{A}^{-1}}$ on ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$. As applications, some maximal inequalities associated with $A$ in the scale of ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ are obtained.

Assuming $T_{0}$ to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$ and prove stability of square root domains, that is,

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2}).\end{eqnarray}$$

$\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})$

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})=\text{dom}(((T_{0}+W)^{\ast })^{1/2}),\end{eqnarray}$$

$$\begin{eqnarray}-\text{div}(a{\rm\nabla}\,\cdot )+(\vec{B}_{1}\cdot {\rm\nabla}\,\cdot )+\text{div}(\vec{B}_{2}\,\cdot )+V\end{eqnarray}$$

$L^{2}({\rm\Omega})$

${\rm\Omega}\subseteq \mathbb{R}^{n}$

$n\in \mathbb{N}$

$\partial {\rm\Omega}$

$\partial {\rm\Omega}$

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’s H-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.