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In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ isthe 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 describethe asymptotic behavior of the Krein spectral shift function (SSF) $\xi(E;H,H_0)$ as the energy $E$ approaches the Landau levels $2bq$, $q \in {\mathbbZ}_+$, which play the role of thresholds in the spectrum of $H_0$. The mainasymptotic term of $\xi(E; H,H_0)$ as $E \to 2bq$ with a fixed $q \in {\mathbbZ}_+$ is written in the terms of appropriate compact Berezin-Toeplitzoperators. Further, we investigate the relation between the thresholdsingularities of the SSF and the accumulation of resonances at the Landaulevels. We establish the existence of resonance free sectors adjoining anygiven Landau level and prove that the number of the resonances in thecomplementary sectors is infinite. Finally, we obtain the main asymptotic termof the local resonance counting function near an arbitrary fixed Landau level;this main asymptotic term is again expressed via the Berezin-Toeplitz operatorswhich govern the asymptotics of the SSF at the Landau levels.

Resonances and Spectral Shift Function Singularities for Magnetic Quantum Hamiltonians