Isoperimetric inequalities for the eigenvalues of natural Schr\"odinger operators on surfaces

TL;DR

This paper investigates eigenvalue optimization for Schr"odinger operators with curvature-dependent potentials on surfaces, showing that spheres maximize the first two eigenvalues under certain conditions.

Contribution

It establishes isoperimetric inequalities for eigenvalues of natural Schr"odinger operators on surfaces, identifying spheres as maximizers for specific eigenvalues.

Findings

Spheres maximize the first eigenvalue of the operator.

Spheres maximize the second eigenvalue under certain conditions.

Results apply to operators with quadratic curvature potentials.

Abstract

This paper deals with eigenvalue optimization problems for a family of natural Schr\"odinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.