Schr\"odinger operators generated by substitutions

TL;DR

This paper reviews recent advances in understanding the spectral properties of Schrödinger operators with potentials generated by primitive substitutions, highlighting conditions for singular continuous spectra and Cantor set support.

Contribution

It provides an overview of new results on spectral characteristics of substitution-generated Schrödinger operators, including verifiable conditions for singular continuous spectra.

Findings

Spectra can be singular continuous under certain conditions

Spectra are supported on Cantor sets of zero Lebesgue measure

Applications to specific substitution examples are discussed

Abstract

Schr\"odinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically verifiable sufficient condition for their spectrum to be singular continuous and supported on a Cantor set of zero Lebesgue measure. Applications to specific examples are discussed.