Schr\"{o}dinger operators with decaying randomness - Pure point spectrum

Abstract

Here we show that for Schr\"{o}dinger 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 Schr\"{o}dinger operator $H^\omega=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_n\omega_n\chi_{_{(0,1]^d}}(x-n)$ 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 $\frac{d\mu}{dx}(x)=O\big(|x|^{-(1+\delta)}\big)~as~|x|\to\infty$, for some $\delta>0$. We show that $H^\omega$ exhibits exponential localization on negative part of spectrum independent of the parameters chosen. For $\alpha\delta\leq d$ 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.