Related papers: Sharp Spectral Asymptotics for two-dimensional Sch…
We consider the spectral problem for the two-dimensional Schr\"odinger operator for a charged particle in strong uniform magnetic and periodic electric fields. The related classical problem is analyzed first by means of the…
We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily…
We consider discrete Schr\"odinger operators $H_{\mu Q}=\Delta+\mu Q$ with real periodic potentials $Q$ on periodic graphs, where $\Delta$ is the adjacency operator and $\mu\in\mathbb R$ is a coupling constant. The spectra of the operators…
In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there…
In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative…
This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal…
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 prove asymptotic formulas of Szeg\H{o} type for the periodic Schr\"odinger operator $H=-\frac{d^2}{dx^2}+V$ in dimension one. Admitting fairly general functions $h$ with $h(0)=0$, we study the trace of the operator…
This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in \cite{Fr,…
We consider a self-adjoint two-dimensional Schr\"odinger operator $H_{\alpha\mu}$, which corresponds to the formal differential expression \[ -\Delta - \alpha\mu, \] where $\mu$ is a finite compactly supported positive Radon measure on…
We consider the magnetic Schr\"odinger operator $H=(i \nabla +A)^2- V$ with a non-negative potential $V$ supported over a strip which is a local deformation of a straight one, and the magnetic field $B:=\mathrm{rot}(A)$ is assumed to be…
We consider the Schr\"odinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the…
We consider a periodic magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a noncompact Riemannian manifold $M$ such that $H^1(M, {\mathbb R})=0$ endowed with a properly discontinuous cocompact…
We consider a magnetic Schr\"odinger operator $H^h=(-ih\nabla-\vec{A})^2$ with the Dirichlet boundary conditions in an open set $\Omega \subset {\mathbb R}^3$, where $h>0$ is a small parameter. We suppose that the minimal value $b_0$ of the…
We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of…
In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…
We study two-dimensional magnetic Schr\"odinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schr\"odinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant…
Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal…
We provide a leading order semiclassical asymptotics of the energy of bound states for magnetic Neumann Schr\"odinger operators in two dimensional (exterior) domains with smooth boundaries. The asymptotics is valid all the way up to the…
We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $\delta'$ type characterized by the coupling parameter $\beta>0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of…