Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

Abstract

In this paper, we consider Schrödinger operators on Mtimes {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 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 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+ieta }(x,y) uniformly in eta >0 . For all energies Ein {mathcal {E}} we prove that the Green’s function G_{E+ieta } is not (uniformly) in ell ^{1} as eta approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.