Spectral pollution in substitution systems

TL;DR

This paper investigates how spectral pollution affects Schrödinger operators in higher-dimensional substitution systems, revealing significant differences from one-dimensional cases due to structural defects.

Contribution

It demonstrates that periodic approximations in higher dimensions can cause spectral pollution, altering essential spectra and measure, unlike in one-dimensional systems.

Findings

Spectral pollution can significantly alter the essential spectrum in higher dimensions.

Structural defects influence the limiting spectral behavior.

Higher-dimensional systems exhibit spectral phenomena not present in one-dimensional cases.

Abstract

We study spectral properties of Schr\"odinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, we analyze how structural defects influence the limiting spectral behavior. In contrast to the one-dimensional setting, we show that such approximations may exhibit significant spectral pollution, including changes in the essential spectrum and the Lebesgue measure.