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

Jordi-Lluís Figueras, Joaquim Puig

TL;DR

This work develops and applies two computer-assisted strategies to prove and bound spectral gaps of the Almost Mathieu Operator at critical coupling $|b|=2$ for irrational frequencies. The dynamical method certifies uniform hyperbolicity of the associated Schrödinger cocycle inside a gap, while the spectral method analyzes rational-frequency periodic approximants and transfers results to irrationals via continuity bounds. For $ω_1=\frac{\sqrt{5}-1}{2}$, eight gaps are rigorously opened; for $ω_2= e-2$, twelve gaps are opened by the spectral approach, with validated endpoint bounds and gap-size statistics. The results provide realistic, rigorously validated gap sizes and spectrum-measure bounds, illustrating the viability of validated numerics in quasi-periodic spectral problems and offering insights into gap distribution, Thouless scaling, and the limits of gap-open questions in the critical regime.

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 $ω=(\sqrt{5}-1)/2$ and 12 of them are open when $ω=e-2$. 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.

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

TL;DR

This work develops and applies two computer-assisted strategies to prove and bound spectral gaps of the Almost Mathieu Operator at critical coupling for irrational frequencies. The dynamical method certifies uniform hyperbolicity of the associated Schrödinger cocycle inside a gap, while the spectral method analyzes rational-frequency periodic approximants and transfers results to irrationals via continuity bounds. For , eight gaps are rigorously opened; for , twelve gaps are opened by the spectral approach, with validated endpoint bounds and gap-size statistics. The results provide realistic, rigorously validated gap sizes and spectrum-measure bounds, illustrating the viability of validated numerics in quasi-periodic spectral problems and offering insights into gap distribution, Thouless scaling, and the limits of gap-open questions in the critical regime.

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 and 12 of them are open when . 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.

Paper Structure

This paper contains 24 sections, 4 theorems, 33 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Dynamics and Gaps of Quasi-Periodic Schrödinger operators
  4. Quasi-periodic cocycles and their reducibility
  5. The rotation number and the Gap Labelling Theorem
  6. Dynamics inside a spectral gap
  7. Dynamical Method: Validation of Uniform Hyperbolicity
  8. Computation of the approximations
  9. Interval arithmetic and validation of solutions
  10. Spectral Method: Periodic approximants
  11. Persistence of gaps. Theoretical results
  12. Almost Mathieu operators with rational frequencies
  13. Computer-assisted proof of the persistence of gaps for irrational frequencies
  14. Results
  15. Dynamical Method
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $(A,\omega)$ be a continuous cocycle on $SL(2,\mathbb{R})$ (for example the Schrödinger cocycle of the AMO) and a frequency $\omega \in \mathbb{R}$. Assume that there exist continuous maps $P_1, P_2: \mathbb{T}\to SL(2,\mathbb{R})$, two constants $\Lambda_{11},\Lambda_{22}\in\mathbb R$, satisfyi Then, if $\sigma+\lambda+\tau < 1$ we obtain that

Figures (7)

  • Figure 1: Hofstadter butterfly: frequency on the vertical direction, spectrum on the horizontal direction.
  • Figure 2: Length of gaps for $p/q=987/1597$ (top) and $p/q = 719/1001$ (bottom) as a function of the absolute value of its label, $|k|$. In red, we plot the cummulative minima of gap length as a function of the absolute value of the label. Note the logarithmic scale on the vertical axis.
  • Figure 3: Validated gaps (horizontal direction) in $\Sigma_{p/q}$ as a function of rational frequencies $p/q$ (vertical direction) with odd $q$, $p$ and $q$ coprime and $q\le 95$.
  • Figure 4: Top: Minimal Gap length for all rational frequencies for $p/q$ with odd $q$, $p$ and $q$ coprime and $q\le 95$. Note the logarithmic scale in the vertical axis. Bottom: the same plot as a function of $p/q$.
  • Figure 5: Measure of the AMO for rational frequencies as a function of the rational frequency $p/q$ for odd $q$ and $q\le 95$ (top) and the same with logarithmic y scale. Greylines connect dots whose frequency $p/q$ have the same $p$. It ressembles of Thomae's function, although the vertical values, the measures $\Sigma_{p,q}$ are not exactly the same for all values $p/q$ with the same $q$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Proposition 2.2: cf Choi1990
  • proof
  • Proposition 2.3: cf Avron1990a
  • Theorem 2.4: cf bellissard-simon
  • Conjecture 3.1
  • Conjecture 3.2