Eigenvalues of the Neumann magnetic Laplacian in the unit disk

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.

33 pages, 3 figures, 4 tables. Minor corrections with respect to version 3. Published in the "Journal of Mathematical Physics"

References in corpus (3)

• Lack of diamagnetism and the Little-Parks effect
• A reverse Faber-Krahn inequality for the magnetic Laplacian
• Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field on disks in the strong field limit