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

Abstract

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $\Omega_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $\xi(E;H_\pm,H_0)$, $E \geq 0$, 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 terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $\Omega_{\rm in}$ onto the plane perpendicular to $B$.