Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

TL;DR

This paper investigates measures with finite local complexity and their impact on the spectral properties of one-dimensional continuum models of quasicrystals, demonstrating conditions for purely singular continuous spectra.

Contribution

It introduces new criteria for measures with finite local complexity and applies Kotani-Remling theory to analyze spectral types of perturbed Laplacians in quasicrystal models.

Findings

Operators have empty absolutely continuous spectrum if measures are non-periodic

Purely singular continuous spectrum is proven for certain continuum quasicrystal models

Finite local complexity measures influence spectral properties significantly

Abstract

We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.