Notes on Lieb-Thirring type inequality for a complex perturbation of fractional Schr\"odinger operator

TL;DR

This paper establishes Lieb-Thirring type inequalities for fractional Schr"odinger operators with complex potentials, extending spectral bounds to non-self-adjoint cases using advanced complex analysis techniques.

Contribution

It introduces new Lieb-Thirring inequalities for fractional Schr"odinger operators with complex potentials, broadening the scope of spectral estimates beyond real-valued potentials.

Findings

Derived Lieb-Thirring inequalities for complex potentials

Extended spectral bounds to fractional Schr"odinger operators

Utilized complex analysis methods from Borichev-Golinskii-Kupin and Hansmann

Abstract

For $s\textgreater{}0$, let $H\_0=(-\Delta)^s$ be the fractional Laplacian. In this paper, we obtainLieb-Thirring type inequalities for the fractional Schr\"odinger operator defined as $H=H\_0+V$,where $V \in L^p(\mathbb{R}^d), p\ge 1, d\ge 1,$ is a complex-valued potential.Our methods are based on results of articles by Borichev-Golinskii-Kupin \cite{BoGoKu} and Hansmann \cite{Ha1}.