Resonances and SSF Singularities for Magnetic Schrödinger Operators

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.