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

We consider the Schrodinger operator $$H_0$$ with constant magnetic field B of scalar intensity $$b>0$$, self-adjoint in $$L^2({{\mathbb {R}}}^3)$$, and its perturbations $$H_+$$ (resp., $$H_-$$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $$\Omega _{\mathrm{in}} \subset {{\mathbb {R}}}^3$$. We introduce the Krein spectral shift functions $$\xi (E;H_\pm ,H_0)$$, $$E \ge 0$$, for the operator pairs $$(H_\pm ,H_0)$$ and study their singularities at the Landau levels $$\Lambda _q : = b(2q+1)$$, $$q \in {{\mathbb {Z}}}_+$$, which play the role of thresholds in the spectrum of $$H_0$$. We show that $$\xi (E;H_+,H_0)$$ remains bounded as $$E \uparrow \Lambda _q$$, $$q \in {{\mathbb {Z}}}_+$$, being fixed, and obtain three asymptotic terms of $$\xi (E;H_-,H_0)$$ as $$E \uparrow \Lambda _q$$, and of $$\xi (E;H_\pm ,H_0)$$ as $$E \downarrow \Lambda _q$$. The first two divergent terms are independent of the perturbation, while the third one involves the logarithmic capacity of the projection of $$\Omega _{\mathrm{in}}$$ onto the plane perpendicular to B.