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

TL;DR

This paper investigates the behavior of the spectral shift function near Landau levels for magnetic Schrödinger operators with boundary perturbations, revealing boundedness and asymptotic behaviors influenced by geometric capacity.

Contribution

It provides a detailed analysis of the singularities of the spectral shift function at Landau levels for magnetic Hamiltonians with boundary conditions, including new asymptotic formulas involving geometric capacity.

Findings

Spectral shift function remains bounded at Landau levels for Dirichlet perturbations.

Asymptotic formulas for the spectral shift function near Landau levels are derived.

The third asymptotic term involves the logarithmic capacity of the boundary projection.

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 $B$.