Eigenvalue bounds for Schr\"odinger operators with complex potentials

TL;DR

This paper establishes bounds on the eigenvalues of Schrödinger operators with complex potentials using L_p-norms, extending previous inequalities to higher dimensions and confirming a conjecture.

Contribution

It extends eigenvalue bounds for Schrödinger operators with complex potentials to higher dimensions, confirming a conjecture and utilizing uniform Sobolev inequalities.

Findings

Bounded the absolute values of eigenvalues using L_p-norms.

Extended previous inequalities to higher dimensions.

Proved a conjecture by Laptev and Safronov.

Abstract

We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher dimensions and proves a conjecture by Laptev and Safronov. Our main ingredient are the uniform Sobolev inequalities of Kenig, Ruiz, and Sogge.