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arXiv · 2411.11721

Eigenvalues of the Neumann magnetic Laplacian in the unit disk

Bernard Helffer·Corentin Léna
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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.

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Bernard Helffer, Corentin Léna. 2024-11-18. Eigenvalues of the Neumann magnetic Laplacian in the unit disk. https://doi.org/10.1063/5.0260068

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