A note on the magnetic Steklov operator on functions

TL;DR

This paper studies the magnetic Steklov eigenvalue problem on manifolds, providing spectral characterizations, bounds, explicit spectrum calculations for certain cases, and a comparison theorem extending classical non-magnetic results.

Contribution

It introduces new characterizations of magnetic Steklov operators, establishes bounds for eigenvalues, computes spectra for specific geometries, and generalizes existing comparison results to magnetic settings.

Findings

Equivalent characterizations of magnetic Steklov operators.

Cheeger-Jammes type lower bound for the first eigenvalue.

Explicit spectrum computation for Euclidean 2- and 4-balls.

Abstract

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean $2$-ball and $4$-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.