Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian

TL;DR

This paper establishes inequalities between magnetic Neumann and Dirichlet eigenvalues for convex domains in 2D and 3D, extending classical results to the magnetic setting with specific geometric considerations.

Contribution

It proves new inequalities relating magnetic Neumann and Dirichlet eigenvalues for convex domains in both two and three dimensions, including rotationally symmetric cases.

Findings

In 2D, the (k+1)-th magnetic Neumann eigenvalue is not larger than the k-th magnetic Dirichlet eigenvalue.

In 3D, for rotationally symmetric convex domains, the (k+2)-th magnetic Neumann eigenvalue is not larger than the k-th magnetic Dirichlet eigenvalue when the latter is simple.

The proofs adapt the Levine-Weinberger strategy to the magnetic Laplacian context.

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. 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 due to Levine and Weinberger.