For the unit disk with a constant magnetic field of strength 2b, the ground state energy of the magnetic Dirichlet-to-Neumann operator satisfies λ_DN(b)=α√b-(α²+2)/6+O(b^{-1/2}) with α=0.76495..., the unique negative zero of the parabolic cylinder function D_{1/2}.
A reverse Faber-Krahn inequality for the magnetic Laplacian
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We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).
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On the magnetic Dirichlet to Neumann operator on the disk -- strong diamagnetism and strong magnetic field limit--
For the unit disk with a constant magnetic field of strength 2b, the ground state energy of the magnetic Dirichlet-to-Neumann operator satisfies λ_DN(b)=α√b-(α²+2)/6+O(b^{-1/2}) with α=0.76495..., the unique negative zero of the parabolic cylinder function D_{1/2}.