Spectra of Anderson type models with decaying randomness

TL;DR

This paper investigates Anderson models with decaying randomness and long-range tails, demonstrating the presence of absolutely continuous and pure point spectra under certain decay conditions.

Contribution

It introduces a class of Anderson models with anisotropic decaying randomness and long-range interactions, analyzing their spectral properties.

Findings

Presence of pure absolutely continuous spectrum

Existence of some pure point spectrum

Spectral properties depend on decay rates

Abstract

In this paper we consider some Anderson type models, with decaying randomness and the free parts having long range tails. The randomness may decay at different rates in different directions, though in majority of directions we require some sort of short range decay. We show that there is pure a.c. spectrum and some pure point spectrum in such models.