Eigenvalue bounds for Dirac and fractional Schr\"odinger operators with complex potentials

TL;DR

This paper establishes Lieb-Thirring-type bounds for fractional Schrödinger and Dirac operators with complex potentials, using resolvent bounds in Schatten spaces to extend spectral analysis techniques.

Contribution

It introduces a novel resolvent bound in Schatten spaces for unperturbed operators, enabling new spectral bounds for complex-valued potentials in these operators.

Findings

Lieb-Thirring-type bounds derived for fractional Schrödinger operators

Spectral bounds established for Dirac operators with complex potentials

Utilization of Schatten space resolvent bounds as a key technique

Abstract

We prove Lieb-Thirring-type bounds for fractional Schr\"odinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin.