On Eigenvalues in Gaps for Perturbed Magnetic Schrr Odinger Operators

1 Introduction (1) We consider Schrr odinger operators with a spectral gap, perturbed by either a decreasing electric potential or a decreasing magnetic eld. The strength of these perturbations depends on a coupling parameter. With growing, eigenvalues may move into the gap or out of the gap. Most of our results concern (lower) bounds for the number of eigenvalues that cross a xed energy level E in the gap, as the coupling parameter increases from 0 to. Using the currently available technology, the rst case (electric perturbations) can be handled in reasonable generality | even if the unperturbed reference operator H contains magnetic terms. In contrast, the second case (magnetic perturbations) resists virtually all attempts to adapt the methods that are so eeective in the rst case. It is one of the aims of the present paper to point out the diierences between these two apparently similar situations. To be more precise, we take for the unperturbed operator H a Schrr odinger operator H = H(~ a; V) = (?ir ?~ a(x)) 2 + V (x) in R n , with a real electric potential V 2 L 1 (R n) and a magnetic potential ~ a 2 C 1 (R n ; R n). Our basic assumption is that H has a gap in the essential spectrum; for instance, this is the case for many periodic potentials V , for some periodic magnetic elds, and also for combinations of electric and magnetic potentials. We also x some energy level E inside the gap. (2) In the rst part of the paper (Sections 1 and 2) we suppose that H is deened as a relatively compact perturbation of H by an electric potential W, where 2 R is a coupling constant and W 2 L 1 (R n) goes to zero at innnity. As grows, some eigenvalue branches of the family H may cross E, and it is a natural question to determine for which perturbations such crossings will occur. This question arises, for instance, in the quantum mechanical theory of impurities in insulators and semiconductors see e.g. the bibliography in DH], H1], GHKSV]. In a second step, one would like to nd the asymptotics of the corresponding counting function ~ N(E; H) := X 0<<<< dimker(H ? E); as ! +1: Here the case of perturbations by non-negative or non-positive electric potentials is suf-ciently well-understood DH], H1], ADH], B1], …