Eigenvalue bounds for Schr\"odinger operators with complex potentials on compact manifolds

TL;DR

This paper establishes optimal eigenvalue bounds for Schr"odinger operators with complex potentials on compact manifolds, extending Euclidean results to curved spaces and demonstrating their sharpness on specific geometries.

Contribution

It introduces eigenvalue bounds for Schr"odinger operators with complex potentials on compact manifolds, generalizing Euclidean results and proving their optimality on Zoll manifolds.

Findings

Eigenvalue bounds depend only on the $L^q$-norm of the potential.

Bounds are shown to be optimal on the round sphere and Zoll manifolds.

Results extend Frank's Euclidean eigenvalue bounds to curved compact manifolds.

Abstract

We prove eigenvalue bounds for Schr\"odinger operator $-\Delta_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the round sphere and more general Zoll manifolds. These bounds are natural analogues of Frank's \cite{MR2820160} results in the Euclidean case.