Improved Lieb-Thirring type inequalities for non-selfadjoint Schr\"odinger operators

TL;DR

This paper enhances Lieb-Thirring inequalities for complex Schrödinger operators, establishing sharper bounds using integrable functions and demonstrating sharpness in one dimension.

Contribution

It provides improved inequalities for non-selfadjoint Schrödinger operators with complex potentials, refining previous bounds and proving sharpness in one dimension.

Findings

Enhanced Lieb-Thirring inequalities for complex potentials

Use of positive, integrable functions in bounds

Sharpness of inequalities in one-dimensional case

Abstract

We improve the Lieb-Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schr\"odinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the one-dimensional case the result is sharp in the sense that if we take a non-integrable function, then an analogous inequality cannot hold.