Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

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 $mathcal{E}subsetSigma_{0}$. 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 $mathbb{Z}^{d}=mathbb{Z}^{d_1}timesmathbb{Z}^{d_2}$. Again, we prove localisation outside $Sigma_{0} $ by showing exponential decay of the Green function $G_{E+ieta}(x,y) $ uniformly in $eta>0 $. For emph{all} energies $Einmathcal{E}$ we prove that the Greens function $G_{E+ieta} $ 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.