Gordon-type arguments in the spectral theory of one-dimensional quasicrystals

TL;DR

This paper reviews recent advances in the spectral theory of one-dimensional quasicrystals, focusing on substitution and circle map potentials, and discusses how local structures influence spectral properties and eigenfunction estimates.

Contribution

It provides a comprehensive overview of recent methods and results in analyzing spectral properties of quasicrystals generated by substitutions and circle maps.

Findings

Absence of eigenvalues in certain spectral regimes

Continuity of spectral measures with respect to Hausdorff measures

Estimates of generalized eigenfunctions based on local repetitive structures

Abstract

We review the recent developments in the spectral theory of discrete one-dimensional Schr\"odinger operators with potentials generated by substitutions and circle maps. We discuss how occurrences of local repetitive structures allow for estimates of generalized eigenfunctions. Among the recent applications of this general approach are almost sure and uniform results on the absence of eigenvalues as well as continuity of the spectral measures with respect to Hausdorff measures.