Related papers: Lowest energy band function for magnetic steps
The effect of a magnetic field on the energy spectrum and on the wave functions of an electron in spherical nano-structures such as single quantum dot and spherical layer is investigated. It is shown that the magnetic field removes the…
We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…
We consider Schr\"odinger operators with periodic electric and magnetic potentials on periodic discrete graphs. The spectrum of such operators consists of an absolutely continuous (a.c.) part (a union of a finite number of non-degenerate…
We show persistence of both Anderson and dynamical localization in Schr\"odinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\"odinger operator with a non-positive ergodic random…
In this paper we investigate a minimization problem related to the principal eigenvalue of the $s$-wave Schr\"{o}dinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the…
We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition…
We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schr\"odinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay…
We consider a periodic Schroedinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional…
In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schr\"odinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to…
We prove that the lowest eigenvalue of the Laplace operator with a magnetic field having a self-intersecting zero set is a monotone function of the parameter defining the strength of the magnetic field, in a neighborhood of infinity. We…
We obtain necessary and sufficient conditions for emerging of small eigenvalue for Schr\"odinger operator on plane under local operator perturbations. In the case the eigenvalue emerges we construct its asymptotics. The examples are given.
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
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…
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…
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…
This note proves convexity resp. concavity of the ground state energy of one dimensional Schr\"odinger operators as a function of an endpoint of the interval for convex resp. concave potentials.
In the 1980s, Helffer and Sj\"ostrand examined in a series of articles the concentration of the ground state of a Schr\"odinger operator in the semiclassical limit. In a similar spirit, and using the asymptotics for the Szeg\"o kernel, we…
We consider a modification of the OMM energy functional which contains an $\ell^1$ penalty term in order to find a sparse representation of the low-lying eigenspace of self-adjoint operators. We analyze the local minima of the modified…
We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…
When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like…