Eigenvalue bounds for the magnetic Laplacian

TL;DR

This paper derives bounds for eigenvalues of the magnetic Laplacian on Riemannian manifolds, providing sharp estimates in specific cases and characterizing equality conditions related to the geometry.

Contribution

It introduces new upper and lower bounds for magnetic Laplacian eigenvalues, including sharp bounds for the first eigenvalue on certain manifolds and domains.

Findings

Sharp upper bound for the first eigenvalue when the potential is a closed 1-form.

Sharp lower bound for the first eigenvalue on 2D Riemannian cylinders.

Characterization of the equality case as a product metric.

Abstract

We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a family of upper bounds for all the eigenvalues, compatible with the Weyl law. When the potential is a closed 1-form, we get a sharp upper bound for the first eigenvalue. In the second part, we consider only closed potentials, and we establish a sharp lower bound for the first eigenvalue when the manifold is a 2-dimensional Riemannian cylinder. The equality case characterizes the situation where the metric is a product. We also look at the case of doubly convex domains in the Euclidean plane.