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.
Categories: math-ph math.MP math.SP