Computer Validation of Open Gaps for the Almost Mathieu Operator with Critical Coupling

TL;DR

This paper introduces computer-assisted techniques to rigorously prove the existence and estimate the size of spectral gaps in the Almost Mathieu operator at critical coupling, confirming predictions for specific frequencies.

Contribution

It develops dynamical and spectral methods for computer-assisted proofs of spectral gaps and provides rigorous numerical estimates for their sizes at critical coupling.

Findings

First 8 gaps are open for omega=(5-1)/2

12 gaps are open for omega=e-2

Conjectures on gap sizes for periodic problems

Abstract

We present some computer assisted methods to prove the existence of spectral gaps for the Almost Mathieu operator at critical coupling and give rigorous numerical estimates on their size. As an example we show that the first 8 gaps predicted by the Gap Labelling theorem are open when $\omega=(\sqrt{5}-1)/2$ and 12 of them are open when $\omega=e-2$. A dynamical method based on the constructive conjugation to a hyperbolic cocycle and a spectral method based on the rigorous computation of the eigenvalues of finite-dimensional matrices are presented. We also present some experiments and conjectures on gap size for the associated peridodic problems.