On non-selfadjoint perturbations of infinite band Schr\"odinger operators and Kato method

TL;DR

This paper establishes Lieb--Thirring inequalities for the discrete spectrum of non-selfadjoint perturbations of multidimensional Schrödinger operators with infinite band spectrum, using Kato's method.

Contribution

It introduces Lieb--Thirring type bounds for non-selfadjoint perturbations of Schrödinger operators with infinite bands, extending previous results to this broader setting.

Findings

Proved Lieb--Thirring inequalities for the discrete spectrum.

Applicable to operators with potentials in L^p spaces.

Extended bounds to non-selfadjoint perturbations with infinite band spectrum.

Abstract

Let $ H_0=-\dd+V_0 $ be a multidimensional Schr\"odinger ope\-rator with a real-valued potential and infinite band spectrum, and $H=H_0+V$ be its non-selfadjoint perturbation defined with the help of Kato approach. We prove Lieb--Thirring type inequalities for the discrete spectrum of $H$ in the case when $V_0\in L^\infty(\br^d)$ and $V\in L^p(\br^d)$, $p>\max(d/2, 1)$.