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Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

In this paper we consider Schrödinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$Γ$-trimmed' random potential, namely a potential which vanishes outside a subset $Γ$ 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 $Σ_{0}=σ(H_{0,Γ^{c}})$ where $H_{0,Γ^{c}} $ is the free (discrete) Laplacian on the complement $Γ^{c} $ of $Γ$. We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subsetΣ_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=\infty$, i.~e.~ $Γ$-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$. Again, we prove localisation outside $Σ_{0} $ by showing exponential decay of the Green function $G_{E+iη}(x,y) $ uniformly in $η>0 $. For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+iη} $ is \emph{not} (uniformly) in $\ell^{1}$ as $η$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here.