Threshold Singularities of the Spectral Shift Function for a Half-Plane Magnetic Hamiltonian

TL;DR

This paper investigates the behavior of the spectral shift function near Landau levels for a magnetic Schrödinger operator on a half-plane, revealing singularities and asymptotics influenced by decaying electric perturbations.

Contribution

It provides a detailed analysis of spectral density near thresholds for magnetic Hamiltonians with boundary conditions, including asymptotic behavior and singularities caused by perturbations.

Findings

Spectral shift function exhibits singularities at Landau levels.

Asymptotic behavior of SSF determined for power-like decaying potentials.

Results extend to Neumann boundary conditions.

Abstract

We consider the Schr\"odinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, $H_0$, and a decaying electric perturbation $V$. We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of $H_0,$ by studying the Spectral Shift Function (SSF) associated to the pair $(H_0+V,{H_0})$. For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of $V$ is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.