Some Estimates Regarding Integrated density of States for Random Schr\"{o}dinger Operator with decaying Random Potentials

TL;DR

This paper studies bounds on the density of states for a class of random Schrödinger operators with decaying potentials, focusing on the pure point spectrum regime, and provides new estimates relevant for understanding spectral properties.

Contribution

It introduces bounds for the density of states in the pure point regime for operators with decaying random potentials, extending previous results to this specific setting.

Findings

Derived bounds for the density of states

Focused on operators with potentials decaying as |n|^{-eta}

Enhanced understanding of spectral properties in the pure point regime

Abstract

We investigate some bounds for the density of states in the pure point regime for the random Schr\"{o}dinger operators $H^{\omega}=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nq_n(\omega)$, acting on $\ell^2(\mathbb{Z}^d)$, where $\{q_n\}$ are iid random variables and $a_n\simeq|n|^{-\alpha}~~\alpha>0$.