Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

TL;DR

This paper investigates quantum wave guides with random potentials, demonstrating localization and delocalization phenomena, the existence of mobility edges, and the limitations of certain analytical methods in these models.

Contribution

It establishes conditions for pure point and absolutely continuous spectra in quantum lattice models with trimmed randomness, revealing mobility edges and analyzing the failure of common localization techniques.

Findings

Localization outside a specific spectral set

Existence of absolutely continuous spectrum in certain regions

Green's function decay properties and methodological limitations

Abstract

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.