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Twenty dry Martinis for the Unitary Almost Mathieu Operator

Christopher Cedzich, Long Li

TL;DR

The paper proves the Dry Ten Martini Problem for the unitary almost Mathieu operator (UAMO) at Diophantine frequencies in the non-critical regime by combining unitary Aubry–André duality with Avila’s almost reducibility theory for one-frequency SU(1,1) cocycles. The UAMO is formulated as a generalized extended CMV matrix with quasi-periodic Verblunsky coefficients, and its spectral problem is translated into Szegő cocycles in SU(1,1). For $\Phi\in\mathrm{DC}$ and $\lambda_1\neq\lambda_2$, all spectral gaps allowed by Gap Labeling are shown to be open, with the spectrum featuring two copies of the Hofstadter butterfly due to enhanced symmetries. The argument relies on almost reducibility of subcritical cocycles, Eliasson-type reducibility for Diophantine frequencies, and a unitary duality that links gap labels to dual eigenfunctions, ruling out collapsed gaps. This work extends dry-gap results to a CMV/GE-CMV setting and highlights the power of cocycle methods and duality in proving gap-opening phenomena.

Abstract

We solve the Dry Ten Martini Problem for the unitary almost Mathieu operator with Diophantine frequencies in the non-critical regime.

Twenty dry Martinis for the Unitary Almost Mathieu Operator

TL;DR

The paper proves the Dry Ten Martini Problem for the unitary almost Mathieu operator (UAMO) at Diophantine frequencies in the non-critical regime by combining unitary Aubry–André duality with Avila’s almost reducibility theory for one-frequency SU(1,1) cocycles. The UAMO is formulated as a generalized extended CMV matrix with quasi-periodic Verblunsky coefficients, and its spectral problem is translated into Szegő cocycles in SU(1,1). For and , all spectral gaps allowed by Gap Labeling are shown to be open, with the spectrum featuring two copies of the Hofstadter butterfly due to enhanced symmetries. The argument relies on almost reducibility of subcritical cocycles, Eliasson-type reducibility for Diophantine frequencies, and a unitary duality that links gap labels to dual eigenfunctions, ruling out collapsed gaps. This work extends dry-gap results to a CMV/GE-CMV setting and highlights the power of cocycle methods and duality in proving gap-opening phenomena.

Abstract

We solve the Dry Ten Martini Problem for the unitary almost Mathieu operator with Diophantine frequencies in the non-critical regime.

Paper Structure

This paper contains 8 sections, 10 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model and results
  3. Preliminaries
  4. The UAMO as a generalized extended CMV matrix
  5. Cocycle Dynamics
  6. Proofs
  7. Proof of Theorem \ref{['thm.Main1']}
  8. Proof of Theorem \ref{['thm.reverDual']}

Key Result

Theorem 2.1

For $\Phi\in {\mathrm{DC}}$ and $\lambda_1\neq\lambda_2$, all gaps in the spectrum of the unitary almost Mathieu operator $W_{\lambda_1,\lambda_2,\Phi}$ allowed by the Gap Labeling Theorem are open.

Figures (1)

  • Figure 1: The "Hofstadter butterfly" for the UAMO in the subcritical regime with $(\lambda_1,\lambda_2)=(1/\sqrt{2},1/\sqrt{3})$ and denominators up to 70. Clearly, there are two butterflies: for every denominator $q$ there are $2q$ bands instead of just $q$ as for the original butterfly hof76. This is rooted in the symmetries of the system: for every $z\in\Sigma_{\lambda_1,\lambda_2,\Phi}$ also $z^*\in\Sigma_{\lambda_1,\lambda_2,\Phi}$ and $-z\in\Sigma_{\lambda_1,\lambda_2,\Phi}$, compare Remark \ref{['rem:twenty']}.

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6: Aubry-André Duality
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • ...and 10 more