Spectral asymptotics of the Neumann Laplacian with variable magnetic field on a smooth bounded domain in three dimensions

TL;DR

This paper analyzes the asymptotic behavior of the lowest eigenvalues of the Neumann magnetic Laplacian in three-dimensional domains, revealing their semiclassical expansion and simplicity under generic magnetic field conditions.

Contribution

It provides a new semiclassical expansion for the lowest eigenvalues of the Neumann magnetic Laplacian with variable magnetic fields in 3D domains.

Findings

Eigenvalues have a semiclassical expansion

Eigenvalues become simple in the semiclassical limit

Results depend on a generic magnetic field assumption

Abstract

This article is devoted the semiclassical spectral analysis of the Neumann magnetic Laplacian on a smooth bounded domain in three dimensions. Under a generic assumption on the variable magnetic field (involving a localization of the eigenfunctions near the boundary), we establish a semiclassical expansion of the lowest eigenvalues. In particular, we prove that the eigenvalues become simple in the semiclassical limit.