Georgi Raikov (Pontificia Universidad Católica de Chile)

Threshold singularities for magnetic quantum Hamiltonians

Abstract: Let H_0 be the 3D Schroedinger operator with constant magnetic field, V be an electric potential which decays sufficiently fast at infinity, and let H = H_0 + V. First, we consider the asymptotic behaviour of the Krein spectral shift function (SSF) for the operator pair (H,H_0) near the Landau levels which play the role of thresholds in the spectrum of H_0. We show that the SSF has singularities near the Landau levels, and describe these singularities in terms of appropriate Berezin - Toeplitz operators. Further, we define the resonances for the operator H and investigate their asymptotic distribution near the Landau levels. We show that under suitable assumptions on the potential V there are infinitely many resonances near every fixed Landau level. We find the main asymptotic term of the corresponding resonance counting function which again is expressed in terms of the Berezin - Toeplitz operators arising in the description of the SSF singularities. If time permits, extensions to magnetic Pauli and Dirac operators will be discussed briefly. The talk is based on joint works with J.-F. Bony (Bordeaux), V. Bruneau (Bordeaux), and C. Fernández (Santiago de Chile).