On the Laplace operator with a weak magnetic field in exterior domains

TL;DR

This paper investigates the spectral properties of the magnetic Laplacian in exterior domains under weak magnetic fields, providing asymptotic eigenvalue estimates and analyzing extremal properties in specific geometries.

Contribution

It offers new asymptotic formulas for eigenvalues in exterior domains and characterizes extremal configurations under magnetic field constraints.

Findings

Asymptotics of low-lying eigenvalues for the exterior of a disk.

Upper bounds on the lowest eigenvalue for star-shaped domains.

The exterior of a disk maximizes the lowest eigenvalue under certain constraints.

Abstract

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the $4$-moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a $p$-moment constraint.