On the semi-classical 3D Neumann Laplacian with variable magnetic field

TL;DR

This paper investigates the semi-classical spectral properties of the 3D Neumann Laplacian with a variable magnetic field, providing new bounds and insights into boundary localization effects.

Contribution

It extends previous work by analyzing the case of variable magnetic fields, offering a three-term upper bound for the lowest eigenvalue and exploring semi-classical spectral behavior.

Findings

Established a three-term upper bound for the lowest eigenvalue.

Demonstrated boundary localization phenomena under variable magnetic fields.

Analyzed semi-classical behavior of the spectrum in three dimensions.

Abstract

In this paper we are interested in the semi-classical estimates of the spectrum of the Neumann Laplacian in dimension 3. This work aims to present a complementary case to the one presented in the paper of Helffer and Morame in the case of constant magnetic field. More precisely, in the case when the magnetic field is variable and under the most generic condition for which boundary localizations can be observed, we prove a three terms upper bound for the lowest eigenvalue and establish some semi-classical behaviour of the spectrum.