Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

TL;DR

This paper numerically investigates the spectral properties of the skew-shift operator, revealing differences from the Almost Mathieu model and supporting conjectures about its spectral gaps and structure.

Contribution

It provides the first comprehensive numerical analysis of the skew-shift operator's spectra, eigenstates, and Lyapunov exponents, supporting conjectures about its spectral gaps.

Findings

Numerical results align with conjectures on spectral gaps.

Spectral features differ significantly from the Almost Mathieu model.

Established a small upper bound on spectral gaps.

Abstract

In this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper's equation. This study is motivated by various conjectures on the spectral theory of these 'pseudo-random' models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).