Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians

Abstract

We consider the 3D Schrödinger 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 ${\Omega }_{\mathrm{in}}\subset {\mathbb{R}}^3$. 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 terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of ${\Omega }_{\mathrm{in}}$ onto the plane perpendicular to $B$.

35 pages

References in corpus (4)

• Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials
• Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains
• Eigenvalue asymptotics of the even-dimensional exterior Landau-Neumann Hamiltonian
• Counting function of magnetic resonances for exterior problems