Related papers: Semiclassical limit for the Schroedinger equation …

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that…

We construct approximate solutions to the time--dependent Schr\"odinger equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V \psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth…

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic…

A periodic one-dimensional Schroedinger operator is called semifinite-gap if every second gap in its spectrum is eventually closed. We construct explicit examples of semifinite-gap Schroedinger operators in trigonometric functions by…

In this paper, we investigate the three-dimensional Schrodinger operator with a periodic, relative to a lattice {\Omega} of R3, potential q. A special class V of the periodic potentials is constructed, which is easily and constructively…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schr\"odinger operators. Schr\"odinger operator $S=-\Delta+V$ is classified as subcritical/critical in terms of the existence/nonexistence of a…

We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this…

This article addresses the microlocalization of eigenfunctions for the semiclassical Schr\"odinger operator $-h^2\Delta+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the…

This paper is a contribution to semiclassical analysis for abstract Schr\"odinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $\mu$ be a Radon measure on $X$ with a full support.…

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an…

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors…

We derive a representation formula for the Weyl solution to the Schr\"odinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with…

We consider a semi-classical Schrodinger operator with a degenerate potential V(x,y) =f(x) g(y) . g is assumed to be a homogeneous positive function of m variables and f is a strictly positive function of n variables, with a strict minimum.…

Starting from the spectrum of Schr\"odinger operators on $\mathbb{R}^n$, we propose a method to detect critical points of the potential. We argue semi-classically on the basis of a mathematically rigorous version of Gutzwiller's trace…

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}{\pi}\int_0^\infty dr\sqrt{-V(r)\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge…

We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential…

We prove that the spectrum of certain non-self-adjoint Schrodinger operators is unstable in the semi-classical limit. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate…

We consider the semiclassical Schr\"odinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2} \Delta + W_l(x) \right)\psi + V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form…

We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$…