相关论文: Semiclassical asymptotics and gaps in the spectra …

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schr\"odinger operator with magnetic wells on a noncompact Riemannian manifold $M$ such that $H^1(M, \R)=0$ equipped with a properly…

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…

We continue our study of a magnetic Schr\"odinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic…

In this paper, we study a refined L2 version of the semiclassical approximation of projectively invariant elliptic operators with invariant Morse type potentials on covering spaces of compact manifolds. We work on the level of spectral…

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

We consider a periodic magnetic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no…

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…

The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like…

A periodic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions…

We consider semiclassical Schr\"odinger operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$. For these operators we establish a sharp spectral asymptotics without full regularity. For the counting function we assume the potential is…

We consider operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$ that locally behave as a magnetic Schr\"odinger operator. For the magnetic Schr\"odinger operators we suppose the magnetic potentials are smooth and the electric potential is…

In this article we study the semiclassical spectral measures associated with Schr\"odinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give…

In our recent papers, we studied semiclassical spectral problems for the Bochner-Schr\"odinger operator on a manifold of bounded geometry. We survey some results of these papers in the setting of the magnetic Schr\"odinger operator in the…

This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L^2-space). For perturbations by a (nonnegative)…

We define a Schr\"odinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the…

We study the asymptotic behavior, in a ``semi-classical limit'', of the first eigenvalues (i.e. the groundstate energies) of a class of Schr\"{o}dinger operators with magnetic fields and the relationship of this behavior with compactness in…

We study the asymptotic behavior as |x| \to \infty of Schr\"odinger operators with homogeneous potentials. For this purpose, we use methods from semiclassical analysis and investigate semiclassical defect mesures. We prove their…