Spectral Approximation for Quasiperiodic Jacobi Operators

TL;DR

This paper introduces an efficient numerical method to approximate the spectra of quasiperiodic Jacobi operators, which model quasicrystals and exhibit complex spectral properties, enabling detailed analysis of aperiodic structures.

Contribution

A simple $O(K^2)$ algorithm for computing spectra of period-$K$ Jacobi operators, improving accuracy and efficiency in spectral analysis of quasiperiodic systems.

Findings

Efficient spectral computation for large periods

Spectral properties of Fibonacci, period doubling, Thue-Morse potentials analyzed

Enhanced understanding of aperiodic order in quasicrystals

Abstract

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into associated dynamical systems. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary to get detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-$K$ Jacobi operator in $O(K^2)$ operations, and use it to investigate the spectra of Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse potentials.