Schrödinger operators with decaying randomness - Pure point spectrum

Abstract

Here we show that 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 operator $H^{\omega }=-\Delta +\sum _{n\in {\mathbb{Z}}^d}{a}_n{\omega }_n{\chi }_{{}_{(0,1{]}^d}}(x-n)$ on $L^2({\mathbb{R}}^d)$. Here we take $a_n=O(|n{|}^{-\alpha })$ for large $n$ where $\alpha >0$, and $\{{\omega }_n{\}}_{n\in {\mathbb{Z}}^d}$ are i.i.d real random variables with absolutely continuous distribution $\mu$ such that $\frac{d\mu }{dx}(x)=O(|x{|}^{-(1+\delta )})as|x|\to \infty$, for some $\delta >0$. We show that $H^{\omega }$ exhibits exponential localization on negative part of spectrum independent of the parameters chosen. For $\alpha \delta \le d$ we show that the spectrum is entire real line almost surely, but for $\alpha \delta >d$ we have ${\sigma }_{ess}(H^{\omega })=[0,\infty )$ and negative part of the spectrum is discrete almost surely. In some cases we show the existence of the absolutely continuous spectrum.

20 pages