Analysis shows the first eigenvalue behavior under Neumann boundary conditions in the unit disk, suggesting implications for magnetic fields.
In this paper, we study the first eigenvalue of the magnetic Laplacian with Neumann boundary conditions in the unit disk D in R2. There is a rather complete asymptotic analysis when the constant magnetic field tends to +∞ 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 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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Helffer et al. (2025) studied this question.
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