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

Let H := H0 + V and H⊥ := H0,⊥ + V be respectively perturbations of the free Schrödinger operators H0 on L2 ( R2d+1 ) and H0,⊥ on L2 ( R2d ) , d ≥ 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 ofH andH⊥. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.