Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian

Abstract

We consider the magnetic Laplacian with the homogeneous magnetic field in two and three dimensions. We prove that the \((k+1)\)-th magnetic Neumann eigenvalue of a bounded convex planar domain is not larger than its k-th magnetic Dirichlet eigenvalue for all \(k\in {{\mathbb {N}}}\). In three dimensions, we restrict our attention to convex domains, which are invariant under rotation by an angle of \(\pi \) around an axis parallel to the magnetic field. For such domains, we prove that the \((k+2)\)-th magnetic Neumann eigenvalue is not larger than the k-th magnetic Dirichlet eigenvalue provided that this Dirichlet eigenvalue is simple. The proofs rely on a modification of the strategy suggested by Payne and developed further by Levine and Weinberger.