Spectra of Random Operators with absolutely continuous Integrated Density of States

TL;DR

This paper investigates the spectral properties of various random operators, establishing that absolute continuity of the integrated density of states (IDS) implies the spectrum is either empty or of positive measure, with broad applicability.

Contribution

It proves that zero density of states measure on subsets implies these subsets are empty, linking absolute continuity of IDS to the nature of the spectrum across multiple models.

Findings

Absolute continuity of IDS implies spectrum is either empty or of positive measure.

Results apply to Anderson, alloy models, and non-ergodic systems.

Spectral subsets with zero measure are empty.

Abstract

The structure of the spectrum of random operators is studied. It is shown that if the density of states measure of some subsets of the spectrum is zero, then these subsets are empty. In particular follows that absolute continuity of the IDS implies singular spectra of ergodic operators is either empty or of positive measure. Our results apply to Anderson and alloy type models, perturbed Landau Hamiltonians, almost periodic potentials and models which are not ergodic.