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

Let H:=H0+VH:=H0+V and H⊥:=H0,⊥+VH⊥:=H0,⊥+V be respectively perturbations of the unperturbed Schrödinger operators H0H0 on L2(R3)L2(R3) and H0,⊥H0,⊥ on L2(R2)L2(R2) with constant magnetic field of strength b>0b>0, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and H⊥H⊥. 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.