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

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 $\mathbb{R}^2$ $Ω$ with smooth boundary as $B$ tends to infinity. We introduce a "magnetic" curvature mixing the curvature of $\partialΩ$ 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.

33 pages, 1 figure, added details/corrections in Section 3.2 and Section 5.3