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

Diomba Sambou (1)

1 IMB - Institut de Mathématiques de Bordeaux

Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ 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.

Mots clés: Magnetic Schrödinger operators; Lieb-Thirring type inequalities; non self-adjoint relatively compact perturbations.; non self-adjoint relatively compact perturbations

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Diomba Sambou. Lieb-Thirring type inequalities for non self-adjoint perturbations of magnetic Schrödinger operators. 2013. ⟨hal-00779606v3⟩