Resonances and Spectral Shift Function Singularities for Magnetic Quantum Hamiltonians

In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schr\"odinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) $\xi(E; H,H_0)$ as the energy $E$ approaches the Landau levels $2bq$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. The main asymptotic term of $\xi(E; H,H_0)$ as $E \to 2bq$ with a fixed $q \in {\mathbb Z}_+$ is written in the terms of appropriate compact Berezin-Toeplitz operators. Further, we investigate the relation between the threshold singularities of the SSF and the accumulation of resonances at the Landau levels. We establish the existence of resonance free sectors adjoining any given Landau level and prove that the number of the resonances in the complementary sectors is infinite. Finally, we obtain the main asymptotic term of the local resonance counting function near an arbitrary fixed Landau level; this main asymptotic term is again expressed via the Berezin-Toeplitz operators which govern the asymptotics of the SSF at the Landau levels.