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Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

, Werner Kirsch
Published in Birkhauser

In this paper, we consider Schrödinger operators on M×Zd2M×Zd2, with M={M1,…,M2}d1M={M1,,M2}d1 (‘quantum wave guides’) with a ‘ΓΓ-trimmed’ random potential, namely a potential which vanishes outside a subset ΓΓ 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 Σ0=σ(H0,Γc)Σ0=σ(H0,Γc) where H0,ΓcH0,Γc is the free (discrete) Laplacian on the complement ΓcΓc of ΓΓ. We also prove that the operators have some absolutely continuous spectrum in an energy region E⊂Σ0EΣ0. Consequently, there is a mobility edge for such models. We also consider the case −M1=M2=∞M1=M2=, i.e. ΓΓ-trimmed operators on Zd=Zd1×Zd2Zd=Zd1×Zd2. Again, we prove localisation outside Σ0Σ0 by showing exponential decay of the Green function GE+iη(x,y)GE+iη(x,y) uniformly in η>0η>0. For all energies E∈EEE we prove that the Green’s function GE+iηGE+iη is not (uniformly) in ℓ11 as ηη approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.

About the journal
JournalAnnales Henri Poincare
Open AccessYes