Remark on magnetic Schroedinger operators in exterior domains

TL;DR

This paper analyzes the spectral properties of magnetic Schrödinger operators in exterior domains, revealing eigenvalue clustering around Landau levels and providing asymptotic formulas for their accumulation rates.

Contribution

It introduces a detailed spectral analysis of magnetic Schrödinger operators with Robin boundary conditions in exterior domains, highlighting eigenvalue clustering behavior.

Findings

Eigenvalues form clusters around Landau levels

Eigenvalues accumulate to Landau levels from below

Asymptotic formulas describe accumulation rates

Abstract

We study the Schroedinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which appears to be independent from the boundary condition.