On eigenvalues in gaps for perturbed magnetic Schrödinger operators

We study Schrödinger operators H0 with a gap in the essential spectrum, perturbed by either a decreasing electric potential or a decreasing magnetic field; in both cases the strength of the perturbation is measured by a coupling constant λ⩾0. Here we are mainly interested in the asymptotic behavior (as λ→∞) of certain counting functions for the eigenvalues that are produced by the perturbation inside the spectral gap. The case where we perturb by a potential can be handled using current technology, even if H0 contains a fixed magnetic background. For perturbations by magnetic fields, however, we require rather strong assumptions—like exponential decay of the perturbations—to obtain a lower bound on the counting function. To gain some additional intuition, we use separation of variables in the closely related model of a Schrödinger operator with constant magnetic field in R2, perturbed by a rotationally symmetric magnetic field that decays at infinity.