Quantum Lattice Wave Guides with Randomness  Localisation and Delocalisation
Abstract
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 `$\Gamma$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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.13686
 arXiv:
 arXiv:2006.13686
 Bibcode:
 2020arXiv200613686K
 Keywords:

 Mathematical Physics;
 Mathematics  Functional Analysis;
 82B44;
 47B80;
 60H25