Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

Abstract

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