Absolutely continuous spectrum and spectral transition for some continuous random operators

TL;DR

This paper investigates the spectral properties of certain continuous random operators, demonstrating the presence of absolutely continuous spectrum and spectral transitions, which are relevant for understanding phenomena like Anderson localization.

Contribution

It introduces two classes of random Hamiltonians on $L^2( r^d)$ with a.c. spectrum and analyzes spectral transitions, extending lattice results to continuous models.

Findings

Existence of absolutely continuous spectrum in both models.

Spectral transition phenomena similar to Bethe lattice and Anderson models.

Presence of dense pure point spectrum under certain disorder conditions.

Abstract

In this paper we consider two classes of random Hamiltonians on $L^2(\RR^d)$ one that imitates the lattice case and the other a Schr\"odinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.