Eigenvalues of Schr\"odinger operators with complex surface potentials

TL;DR

This paper investigates the spectral properties of Schr"odinger operators with complex surface potentials, establishing bounds on eigenvalues' locations and sums based on the potential's $L^p$ norm.

Contribution

It provides new bounds on eigenvalues of Schr"odinger operators with complex surface potentials, linking eigenvalue distribution to the potential's $L^p$ norm.

Findings

Eigenvalues lie within a bounded disk in the complex plane.

Bounds on sums of powers of eigenvalues are established.

Eigenvalue location depends on the $L^p$ norm of the potential.

Abstract

We consider Schr\"odinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with $d-1<p\leq d$. We also prove bounds on sums of powers of eigenvalues.