Surface Lifshits tails for random quantum Hamiltonians

We consider Schr\"{o}dinger operators on $L^{2}({\mathbb R}^{d})\otimes L^{2}({\mathbb R}^{\ell})$ of the form $H_{\omega}~=~H_{\perp}\otimes I_{\parallel} + I_{\perp} \otimes {H_\parallel} + V_{\omega}$, where $H_{\perp}$ and $H_{\parallel}$ are Schr\"{o}dinger operators on $L^{2}({\mathbb R}^{d})$ and $L^{2}({\mathbb R}^{\ell})$ respectively, and $V_\omega(x,y)$ : = $\sum_{\xi \in {\mathbb Z}^{d}} \lambda_\xi(\omega) v(x - \xi, y)$, $x \in {\mathbb R}^d$, $y \in {\mathbb R}^\ell$, is a random 'surface potential'. We investigate the behavior of the integrated density of surface states of $H_{\omega}$ near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of $H_{\omega}$ can be read off from the integrated density of states of a reduced Hamiltonian $H_{\perp}+W_{\omega}$ where $W_{\omega}$ is a quantum mechanical average of $V_{\omega}$ with respect to $y \in {\mathbb R}^\ell$. We are particularly interested in cases when $H_{\perp}$ is a magnetic Schr\"{o}dinger operator, but we also recover some of the results from [24] for non-magnetic $H_{\perp}$.