Lieb-Thirring type inequalities for non self-adjoint perturbations of magnetic Schrödinger operators

Abstract

Let $H:=H_0+V$ and $H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the free Schrödinger operators $H_0$ on $L^2\left({\mathbb{R}}^{2d+1}\right)$ and $H_{0,\perp }$ on $L^2\left({\mathbb{R}}^{2d}\right)$, $d\ge 1$ 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 $apriori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.