Lieb-Thirring estimates for non self-adjoint Schr\"odinger operators

TL;DR

This paper establishes bounds on the negative eigenvalues of non-self-adjoint operators using their symmetric parts, leading to Lieb-Thirring estimates for non-self-adjoint Schrödinger operators and insights into resonances.

Contribution

It introduces a method to bound eigenvalue moments of non-symmetric operators via their symmetric parts, extending Lieb-Thirring estimates to non-self-adjoint Schrödinger operators.

Findings

Bound on negative eigenvalues using symmetric part of operator

Lieb-Thirring estimates for non-self-adjoint Schrödinger operators

Discussion on resonances of Schrödinger operators

Abstract

For general non-symmetric operators $A$, we prove that the moment of order $\gamma \ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $\gamma$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schr\"odinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schr\"odinger self-adjoint operators.