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

TL;DR

This paper investigates the spectral properties of Schr"{o}dinger operators with decaying random potentials, revealing a transition from essential to discrete spectrum in the negative part and demonstrating localization and spectrum types.

Contribution

It establishes conditions under which the negative spectrum transitions from essential to discrete, depending on decay rate and tail behavior of the potential distribution.

Findings

Negative spectrum exhibits exponential localization.

Spectrum is entire real line under certain decay conditions.

Existence of absolutely continuous spectrum in some cases.

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.