  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
$H_{\perp}$.
