Spectral approximation for substitution systems

TL;DR

This paper investigates spectral approximations of aperiodic Schrödinger operators derived from symbolic substitution systems, establishing convergence criteria, new examples, and limitations in higher dimensions, with implications for spectral analysis.

Contribution

It characterizes spectral convergence for substitution-based Schrödinger operators on Lie groups and introduces new periodic approximation examples, highlighting differences between one and higher dimensions.

Findings

Spectral convergence in Hausdorff distance is characterized by properties of finite graphs.

New examples of periodic approximations are constructed.

Some substitution systems in higher dimensions do not admit periodic approximations.

Abstract

We study periodic approximations of aperiodic Schr\"odinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schr\"odinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$.