Sharp asymptotics for the Neumann Laplacian with variable magnetic field : case of dimension 2

TL;DR

This paper derives precise asymptotic estimates for the lowest eigenvalue of the Neumann Laplacian with variable magnetic field in two dimensions, incorporating boundary curvature and magnetic field variations.

Contribution

It introduces a magnetic curvature concept and provides sharp eigenvalue estimates for variable magnetic fields with non-degenerate minima on the boundary.

Findings

Eigenvalue estimates analogous to constant magnetic field case

Precise asymptotics when magnetic field has a unique non-degenerate minimum

Estimate of the third critical field in Ginzburg-Landau theory with variable magnetic field

Abstract

The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of $(i\nabla+B\textbf{A})^2$ on an open bounded subset of $\mathbb{R}^2$ $\Omega$ with smooth boundary as $B$ tends to infinity. We introduce a "magnetic" curvature mixing the curvature of $\partial\Omega$ and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg-Landau theory in the variable magnetic field case.