Clusters of eigenvalues for the magnetic Laplacian with Robin condition

TL;DR

This paper analyzes the spectrum of a magnetic Schrödinger operator with Robin boundary conditions outside a compact domain, revealing eigenvalue clusters around Landau levels and providing asymptotic accumulation rates.

Contribution

It introduces a detailed spectral analysis of the magnetic Laplacian with Robin boundary conditions, including precise asymptotics for eigenvalue clustering.

Findings

Eigenvalues form clusters around Landau levels

Eigenvalues accumulate to Landau levels from below

Asymptotic formula for accumulation rate is derived

Abstract

We study the Schr\"{o}dinger operator with a constant magnetic field in the exterior of a compact domain in euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic 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 is independent of the boundary condition.