In this paper we consider Schr\"{ o}dinger operators on $M \times \mathbb{ Z}^{ d_2}$, with $M=\{ M_{ 1}, \ldots, M_{ 2}\}^{ d_1}$ (`quantum wave guides') with a `$\Gamma$-trimmed' random potential, namely a potential which vanishes outside a subset $\Gamma$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have \emph{ pure point spectrum } outside the set $\Sigma_{ 0}=\sigma(H_{ 0,\Gamma^{ c}})$ where $H_{ 0,\Gamma^{ c}} $ is the free (discrete) Laplacian on the complement $\Gamma^{ c} $ of $\Gamma $. We also prove that the operators have some \emph{ absolutely continuous spectrum} in an energy region $\mathcal{ E}\subset\Sigma_{ 0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{ 1}=M_{ 2}=\infty$, i.~e.~ $\Gamma $-trimmed operators on $\mathbb{ Z}^{ d}=\mathbb{ Z}^{ d_1}\times\mathbb{ Z}^{ d_2}$. Again, we prove localisation outside $\Sigma_{ 0} $ by showing exponential decay of the Green function $G_{ E+i\eta}(x,y) $ uniformly in $\eta>0 $. For \emph{ all} energies $E\in\mathcal{ E}$ we prove that the Green's function $G_{ E+i\eta} $ is \emph{ not} (uniformly) in $\ell^{ 1}$ as $\eta$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{ can} be applied here.
Categories: math-ph math.FA math.MP