The IDS and Asymptotic of the Largest Eigenvalue of Random Schr\"{o}dinger Operators with Decaying Random Potential

TL;DR

This paper investigates the spectral properties of Schrödinger operators with decaying random potentials, focusing on the integrated density of states and the asymptotic behavior of extreme eigenvalues in finite volume approximations.

Contribution

It provides new results on the integrated density of states and the asymptotics of the largest eigenvalue for Schrödinger operators with decaying random potentials.

Findings

Derived the integrated density of states for the operator

Analyzed the asymptotic behavior of the largest eigenvalue

Provided insights into spectral properties with decaying randomness

Abstract

In this work we obtain the integrated density of states for the Schr\"{o}dinger operators with decaying random potentials acting on $\ell^2(\mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation