Resonances and SSF Singularities for Magnetic Schrödinger Operators

Jean-François Bony; Vincent Bruneau; Philippe Briet; Georgi Raikov

Authors

Jean-François Bony Université Bordeaux I, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, 351, Cours de la Libération, 33405 Talence, France.
Vincent Bruneau Université Bordeaux I, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, 351, Cours de la Libération, 33405 Talence, France.
Philippe Briet Centre de Physique Théorique, CNRS-Luminy, Case 907, 13288 Marseille, France.
Georgi Raikov Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile.

Keywords:

Magnetic Schrödinger operators, resonances, spectral shift function

Abstract

The aim of this note is to review recent articles on the spectral properties of magnetic Schrödinger operators. We consider H₀, a 3D Schrödinger operator with constant magnetic field, and ˜H₀, a perturbation of H₀ by an electric potential which depends only on the variable along the magnetic field. Let H (resp. ˜H ) be a short range perturbation of H₀ (resp. of ˜H₀). In the case of (H,H₀), we study the local singularities of the Krein spectral shift function (SSF) and the distribution of the resonances of H near the Landau levels which play the role of spectral thresholds. In the case of ( ˜H, ˜H₀), we study similar problems near the eigenvalues
of ˜H₀ of infinite multiplicity.