The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain

TL;DR

This paper investigates the spectral properties of the magnetic Laplacian with a higher-order vanishing magnetic field in bounded domains, analyzing asymptotic behaviors of ground state energies as the magnetic field strength grows large.

Contribution

It provides a unified approach to asymptotic analysis of the magnetic Laplacian's spectrum for Dirichlet, Neumann, and Dirichlet-to-Neumann cases with higher-order vanishing magnetic fields.

Findings

Established leading orders of $eta$ for ground state energies.

Derived first terms in asymptotic expansions with remainder estimates.

Unified analysis for different boundary conditions.

Abstract

This paper is concerned with spectrum properties of the magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. We study the asymptotic behaviors of ground state energies for the Dirichlet Laplacian, the Neumann Laplacian, and the Dirichlet-to-Neumann operator, as the field strength parameter $\beta$ goes to infinite. Assume that the magnetic field does not vanish to infinite order, we establish the leading orders of $\beta$. We also obtain the first terms in the asymptotic expansions with remainder estimates under additional assumptions on an invariant subspace for a Taylor polynomial of the magnetic field. Our aim is to provide a unified approach to all three cases.