Eigenvalues of the Neumann magnetic Laplacian in the unit disk

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.