Related papers: Weyl asymptotics for magnetic Schr\"odinger operat…

We present an analytical investigation of the asymptotic behavior of non-resonance eigenvalues for the fractional Schr\"odinger operator under homogeneous Neumann boundary conditions. Our findings reveal an intriguing convergence: as the…

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schr\"odinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large…

Let $G\subset \O(n)$ be a group of isometries acting on $n$-dimensional Euclidean space $\R^n$, and ${\bf{X}}$ a bounded domain in $\R^n$ which is transformed into itself under the action of G. Consider a symmetric, classical…

We consider a Schr\"odinger operator $(h\mathbf D -\mathbf A)^2$ with a positive magnetic field $B=\curl\mathbf A$ in a domain $\Omega\subset\R^2$. The imposing of Neumann boundary conditions leads to spectrum below $h\inf B$. This is a…

We compute the sum and number of eigenvalues for a certain class of magnetic Schrodinger operators in a domain with boundary. Functions in the domain of the operator satisfy a (magnetic) Robin condition. The calculations are valid in the…

We study the eigenvalues of the non-self adjoint problem $-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a…

In this paper, we consider the 2D- Schr\"odinger operator with constant magnetic field $H(V)=(D_x-By)^2+D_y^2+V_h(x,y)$, where $V$ tends to zero at infinity and $h$ is a small positive parameter. We will be concerned with two cases: the…

This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…

We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field; it is given by Weyl expression with O(h^{-1}) error and under standard condition to Hamiltonian trajectories even…

We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$…

We consider a magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the…

We prove a sharp Weyl estimate for the number of eigenvalues belonging to a fixed interval of energy of a self-adjoint difference operator acting on $\ell^2(\epsilon\mathbb{Z}^d)$ if the associated symplectic volume of phase space in…

Let $G\subset \O(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space $\R^n$, and ${\bf{X}}$ a bounded domain in $\R^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical…

We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schr\"odinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the…

This paper studies the Schr\"odinger operator with Morse potential on a right half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains information on zeros of the Whittaker…

We consider a general second-order elliptic differential operator on a domain with a cylindrical end. We impose Dirichlet boundary conditions on the boundary with the exception of a small set, where we impose Neumann boundary conditions.…

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schr\"odinger operator on $L^2(\bR^n)$ with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the…

We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schr\"odinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value $b_0$ of the intensity of the magnetic field is strictly…

We prove a universal bound for the number of negative eigenvalues of Schr\"odinger operators with Neumann boundary conditions on bounded H\"older domains, under suitable assumptions on the H\"older exponent and the external potential. Our…

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl…