Lieb-Thirring type inequalities for non self-adjoint perturbations of magnetic Schr\"odinger operators

TL;DR

This paper establishes Lieb-Thirring inequalities for the discrete spectra of magnetic Schrödinger operators with complex perturbations, providing bounds on eigenvalue distribution and convergence near Landau levels.

Contribution

It introduces new Lieb-Thirring type bounds for non self-adjoint magnetic Schrödinger operators, advancing understanding of eigenvalue distribution in such systems.

Findings

Derived inequalities for eigenvalue distribution around Landau levels

Provided bounds on the convergence rate of eigenvalues

Extended Lieb-Thirring inequalities to non self-adjoint magnetic operators

Abstract

Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schr\"odinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.