Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

摘要

We consider the Schrodinger operator H0with constant magnetic field B of scalar intensity b>0self-adjoint in L2(R3) and its perturbations H+ (resp., H-obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain omega in subset of R3 We introduce the Krein spectral shift functions xi(Ex37e;H +/-,H0) for the operator pairs (H +/-,H0)and study their singularities at the Landau levels ?q:=b(2q+1)which play the role of thresholds in the spectrum of H0 We show that xi(Ex37e;H+,H0)remains bounded as E up arrow?qbeing fixed, and obtain three asymptotic terms of xi(Ex37e;H-,H0) as E up arrow?q$$E uparrow Lambda _q$$end{document}, and of xi(Ex37e;H +/-,H0)as E down arrow?qThe first two divergent terms are independent of the perturbation, while the third one involves the logarithmic capacity of the projection of omega inonto the plane perpendicular to B.