Eigenvalues of the Neumann magnetic Laplacian in the unit disk

TL;DR

This paper investigates the first eigenvalue of the Neumann magnetic Laplacian in the unit disk, combining classical physics insights with modern asymptotic analysis to improve understanding of its behavior under strong magnetic fields.

Contribution

It revisits classical physics models and enhances asymptotic analysis of the magnetic Laplacian eigenvalues in the disk, providing improved theoretical and numerical insights.

Findings

Asymptotic behavior of eigenvalues for strong magnetic fields

Improved formulas combining Saint-James and Fournais-Helffer results

Numerical validation of theoretical conjectures

Abstract

In this paper, we study the first eigenvalue of the magnetic Laplacian with Neumann boundary conditions in the unit disk $\mathbb D$ in $\mathbb R^2$. There is a rather complete asymptotic analysis when the constant magnetic field tends to $+\infty$ and some inequalities seem to hold for any value of this magnetic field, leading to rather simple conjectures. Our goal is to explore these questions by revisiting a classical picture of the physicist D. Saint-James theoretically and numerically. On the way, we revisit the asymptotic analysis in light of the asymptotics obtained by Fournais-Helffer, that we can improve by combining them with a formula stated by Saint-James.