Lower bounds for the first eigenvalue of the magnetic Laplacian

TL;DR

This paper derives sharp lower bounds for the first eigenvalue of the magnetic Laplacian on Riemannian cylinders and planar domains, linking spectral properties to geometric and topological features.

Contribution

It provides new explicit lower bounds for the magnetic Laplacian's first eigenvalue, characterizing equality cases and extending results to planar domains with boundary.

Findings

Sharp lower bound for the magnetic Laplacian's first eigenvalue on Riemannian cylinders.

Explicit geometric lower bound for planar domains bounded by two closed curves.

Characterization of equality cases where the metric is a product.

Abstract

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.