Estimates for the Lowest Eigenvalue of Magnetic Laplacians

TL;DR

This paper derives bounds for the lowest eigenvalues of magnetic Laplacians on bounded domains, relating them to geometric properties and considering both Dirichlet and Neumann boundary conditions in the presence of constant magnetic fields.

Contribution

It provides new estimates for the first eigenvalues of magnetic Laplacians, including bounds based on domain geometry and results for Neumann conditions under constant magnetic fields.

Findings

Lower and upper bounds for Dirichlet eigenvalues with constant magnetic field

A lower bound for the Neumann eigenvalue in the constant field case

Relations between eigenvalues and geometric quantities of the domain

Abstract

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.