Schrödinger operators with decaying randomness - Pure point spectrum

Abstract

For Schrödinger operator with decaying random potential with fat tail single site distribution the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schrödinger operatorHere we take a n = O(|n| −α ) for large n where α > 0, and {ω n } n∈Z d are i.i.d real random variables with absolutely continuous distribution µ such that dµ dx (x) = O |x| −(1+δ) as |x| → ∞, for some δ > 0. We show that H ω exhibits exponential localization on negative part of spectrum independent of the parameters chosen. For αδ ≤ d we show that the spectrum is entire real line almost surely, but for αδ > d we have σ ess (H ω ) = [0, ∞) and negative part of the spectrum is discrete almost surely. In some cases we show the existence of the absolutely continuous spectrum.