Spectral shift function in strong magnetic fields

We consider the three-dimensional Schrödinger operator $H$ with constant magnetic field of strength $b>0$, and with continuous electric potential $V\in L^1(\mathbb R^3)$ that admits certain power-like estimates at infinity. The asymptotic behavior as $b\to\infty$ of the spectral shift function $\xi(E;H,H_0)$ is studied for the pair of operators $(H,H_0)$ at the energies $\mathcal E=\mathcal{E}b+\lambda$, $\mathcal E>0$ and $\lambda\in\mathbb R$ being fixed. Two asymptotic regimes are distinguished. In the first one, called asymptotics far from the Landau levels, we pick $\mathcal E/2\notin\mathbb Z$ and $\lambda\in\mathbb R$; then the main term is always of order $\sqrt b$, and is independent of $\lambda$. In the second asymptotic regime, called asymptotics near a Landau level, we choose $\mathcal E=2q_0$, $q_o\in\mathbb Z_+$, and $\lambda\ne0$; in this case the leading term of the SSF could be of order $b$ or $\sqrt b$ for different $\lambda$.