Eigenfunction Statistics for Anderson Model with H\"{o}lder continuous single site Potential

TL;DR

This paper investigates the statistical behavior of eigenfunctions for Anderson models with Hölder continuous potentials, demonstrating Poisson distribution limits in the localized regime, advancing understanding of spectral properties in disordered quantum systems.

Contribution

It establishes Poisson limits for eigenfunction distributions in Anderson models with Hölder continuous potentials, extending prior results to a broader class of disorder distributions.

Findings

Eigenfunction distributions converge to Poisson in the localized regime.

Results apply to potentials with Hölder continuity ($0<eta ext{ } extless ext{ }1$).

Provides new insights into spectral statistics of disordered systems.

Abstract

We consider random Schr\"{o}dinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $\alpha$-H\"{o}lder continuous ($0<\alpha\leq 1$). In localized regime we study the distribution of eigenfunctions simultaneously in space and energy. In a certain scaling limit we prove limits point are Poisson.