Abstract

We consider the Schrödinger operator H with constant magnetic field B of scalar intensity b> 0 , self-adjoint in L2(R3) , and its perturbations H+ (resp., H-) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain Ω in⊂ R3. We introduce the Krein spectral shift functions ξ(E; H±, H) , E≥ 0 , for the operator pairs (H±, H) and study their singularities at the Landau levels Λ q: = b(2 q+ 1) , q∈ Z+, which play the role of thresholds in the spectrum of H. We show that ξ(E; H+, H) remains bounded as E↑ Λ q, q∈ Z+, being fixed, and obtain three asymptotic terms of ξ(E; H-, H) as E↑ Λ q, and of ξ(E; H±, H) as E↓ Λ q. The first two divergent terms are independent of the perturbation, while the third one involves the logarithmic capacity of the projection of Ω in onto the plane perpendicular to B.