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

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.