Eigenvalue bounds of the Robin Laplacian with magnetic field

TL;DR

This paper derives bounds for the eigenvalues of the magnetic Laplacian with Robin boundary conditions on a Riemannian manifold, relating them to geometric properties and comparing magnetic and non-magnetic cases.

Contribution

It provides new eigenvalue estimates for the magnetic Laplacian with Robin boundary conditions on manifolds, incorporating magnetic field effects and geometric data.

Findings

Eigenvalue bounds expressed via mean curvature, Robin parameter, and Ricci curvature lower bound.

Comparison of magnetic and non-magnetic eigenvalues using min-max principle.

Use of Bochner formula for magnetic Laplacian to derive estimates.

Abstract

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(\lambda\_k(\tau,\alpha))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $\tau$ is a positive number that defines the Robin condition and $\alpha$ is a real differential 1-form on $M$ that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter $\tau$ and a lower bound of the Ricci curvature of $M$ (see Theorem \ref{estimate1} and Corollary \ref{corestimate}). The main technique is to use the Bochner formula established in \cite{ELMP} for the magnetic Laplacian and to integrate it over $M$ (see Theorem \ref{bochnermagnetic1}). In the last part, we compare the eigenvalues $\lambda\_k(\tau,\alpha)$ with the first eigenvalue $\lambda\_1(\tau)=\lambda\_1(\tau,0)$ (i.e. without magnetic field) and the Neumann eigenvalues $\lambda\_k(0,\alpha)$ (see Theorem \ref{thm:comp}) using the min-max principle.