Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D

TL;DR

This paper classifies 1D aperiodic quantum systems that admit periodic spectral approximations, providing explicit constructions for systems like Fibonacci and Golay-Rudin-Shapiro sequences, using topological and graph-theoretic tools.

Contribution

It offers a complete classification of 1D systems with convergent spectral approximations and introduces explicit construction methods for key examples.

Findings

Complete classification of 1D aperiodic systems with spectral approximations

Explicit constructions for Fibonacci and Golay-Rudin-Shapiro systems

Connection between GAP-graph structures and spectral defects

Abstract

The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schr\"odinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.