Discontinuities of the integrated density of states for random operators on Delone sets

TL;DR

This paper investigates the discontinuities in the integrated density of states for random operators on Delone sets, revealing that such discontinuities are caused by localized eigenfunctions with bounded support, distinguishing quasicrystal models from traditional random models.

Contribution

It identifies the local origin of discontinuities in the integrated density of states for quasicrystal models, highlighting the role of eigenfunctions with bounded support.

Findings

Discontinuities are caused by eigenfunctions with bounded support.

The phenomenon is a local effect specific to quasicrystal models.

This distinguishes quasicrystals from usual random models.

Abstract

Despite all the analogies with "usual random" models, tight binding operators for quasicrystals exhibit a feature which clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to occurrence of eigenfunctions with bounded support.