The Dry Ten Martini Problem asks whether every spectral gap predicted by the Gap Labeling Theorem for a quasiperiodic operator is actually open rather than collapsed. For the unitary almost Mathieu operator (UAMO) — a quasiperiodic split-step quantum walk that is the unitary analogue of the almost Mathieu operator — Cedzich and Li (2025) proved this "all gaps open" property only in the non-critical regime λ1≠λ2\lambda_1\neq\lambda_2 with Diophantine frequency. The problem is to complete the Dry Ten Martini Problem for the UAMO in the remaining regimes: the critical diagonal λ1=λ2\lambda_1=\lambda_2, where the Cantor-spectrum (Ten Martini) property is known for every irrational frequency but no gap-opening statement is known, and the non-critical regime with Liouville (non-Diophantine) frequencies. The problem remained unresolved as of August 2026.

Background

The almost Mathieu operator (AMO) is the one-dimensional quasiperiodic Schrödinger operator (Hλ,Φ,θψ)(n)=ψ(n+1)+ψ(n−1)+2λcos⁡(2π(nΦ+θ))ψ(n)(H_{\lambda,\Phi,\theta}\psi)(n)=\psi(n+1)+\psi(n-1)+2\lambda\cos(2\pi(n\Phi+\theta))\psi(n) on ℓ2(Z)\ell^2(\mathbb{Z}), where λ≥0\lambda\geq0 is the coupling, the frequency Φ\Phi is irrational, and θ\theta is the phase; it models an electron on a lattice in a magnetic field and generates Hofstadter's butterfly. Simon's 1982 review publicized, after Kac, the Ten Martini Problem — is the spectrum of Hλ,Φ,θH_{\lambda,\Phi,\theta} a Cantor set? — which Avila and Jitomirskaya (2009) answered affirmatively for all nontrivial couplings and all irrational frequencies. The Dry Ten Martini Problem (DTMP), posed in the same review, asks for the sharper statement that all spectral gaps allowed by the Gap Labeling Theorem are open: the gap labeling theorem predicts which values kΦ mod 1k\Phi \bmod 1, k∈Zk\in\mathbb{Z}, can label gaps (via the integrated density of states, equivalently the fibered rotation number of the transfer-matrix cocycle), and the "dry" question is whether every predicted label labels a genuine nondegenerate gap rather than a collapsed one.

For the AMO the dry question is nearly settled. Choi, Elliott and Yui (1990) settled the critical coupling with Liouville frequencies via Gauss polynomials and the rotation algebra; Puig (2004) settled the subcritical and Aubry-dual regimes via reducibility of the transfer-matrix cocycle; and Avila, You and Zhou (2023) settled the entire non-critical regime λ≠±1\lambda\neq\pm1 for all irrational frequencies. The sole remaining open case of the classical problem is the critical coupling λ=1\lambda=1 with Diophantine (non-Liouville) frequency, where even computer-assisted work only certifies that the first finitely many gaps open for specific frequencies (Figueras–Puig, 2024).

The unitary almost Mathieu operator, introduced by Cedzich, Fillman and Ong (2023), is the unitary counterpart of the AMO: a discrete-time quantum walk describing a particle with a two-dimensional internal degree of freedom moving on Z\mathbb{Z}, obtained by alternating a parametrized conditional shift Sλ1S_{\lambda_1}, defined by δn±↦λ1δn±1±±1−λ12 δn∓\delta^{\pm}_n\mapsto\lambda_1\delta^{\pm}_{n\pm1}\pm\sqrt{1-\lambda_1^2}\,\delta^{\mp}_n, with a "coin" operator Qλ2Q_{\lambda_2} acting at each site through the modified rotation matrix Qλ2,2π(nΦ+θ)Q_{\lambda_2,2\pi(n\Phi+\theta)} with entries built from λ2cos⁡\lambda_2\cos, λ2sin⁡\lambda_2\sin and ±i1−λ22\pm i\sqrt{1-\lambda_2^2}. The UAMO is Wλ1,λ2,Φ,θ:=Sλ1Qλ2,Φ,θW_{\lambda_1,\lambda_2,\Phi,\theta}:=S_{\lambda_1}Q_{\lambda_2,\Phi,\theta} with couplings λ1,λ2∈[0,1]\lambda_1,\lambda_2\in[0,1]; the frequency Φ\Phi plays the role of a magnetic field and θ\theta the role of a phase. Being unitary, its spectrum is a phase-independent compact subset Σλ1,λ2,Φ\Sigma_{\lambda_1,\lambda_2,\Phi} of the unit circle. Through the theory of CMV matrices — the five-diagonal unitary matrices, determined by a sequence of Verblunsky coefficients, that play for unitary operators the role Jacobi matrices play for Schrödinger operators — the spectral analysis of the UAMO reduces to an SU(1,1)\mathrm{SU}(1,1)-valued Szegő cocycle over the circle rotation θ↦θ+Φ\theta\mapsto\theta+\Phi. Gaps of Σλ1,λ2,Φ\Sigma_{\lambda_1,\lambda_2,\Phi} are open arcs of the unit circle disjoint from the spectrum; the Gap Labeling Theorem assigns to a gap the value of the cocycle's fibered rotation number rot⁡(Φ,Sz)\operatorname{rot}(\Phi,S_z), so the possible labels are kΦ/2 mod 1k\Phi/2 \bmod 1, k∈Zk\in\mathbb{Z} — twice the density of AMO labels, whence the "twenty" dry Martinis of the source paper's title. A label may be realized by an open gap, or be collapsed, when no nondegenerate gap carries it.

Cedzich–Fillman–Ong computed the complete spectral phase diagram using Avila's global theory of one-frequency cocycles: for λ1>λ2\lambda_1>\lambda_2 ("subcritical") the spectrum is purely absolutely continuous; for 0<λ1=λ2≤10<\lambda_1=\lambda_2\leq1 ("critical") it is purely singular continuous and is a zero-measure Cantor set for every irrational Φ\Phi, which is the Ten Martini property in the critical regime; for λ1<λ2\lambda_1<\lambda_2 ("supercritical") the Lyapunov exponent — the asymptotic exponential growth rate of the transfer matrices — is positive and one has Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) for almost every phase. The model also admits an Aubry–André duality, a twisted Fourier transform relating Wλ1,λ2W_{\lambda_1,\lambda_2} to (a transpose of) Wλ2,λ1W_{\lambda_2,\lambda_1}, the unitary analogue of the duality at the heart of the classical AMO theory.

Cedzich and Li (2025) proved the DTMP for the UAMO in the non-critical regime with Diophantine frequencies — Φ∈DC(κ,τ)\Phi\in DC(\kappa,\tau), meaning ∥nΦ∥T≥κ∣n∣−τ\|n\Phi\|_{\mathbb{T}}\geq\kappa|n|^{-\tau} for all n≠0n\neq0, while Liouville frequencies are the well-approximable complement. Their proof combines subcriticality with Avila's almost reducibility theorem (every subcritical analytic cocycle is a limit of conjugates of constant cocycles), Eliasson-type reducibility at Diophantine frequency, Puig's perturbative gap-opening scheme, Aubry–André duality, and an arithmetic Anderson localization theorem valid for Φ\Phi-nonresonant phases. Remark 2.3 of that paper states explicitly what the method cannot reach: the dry version in the critical regime "remains open" (only the Cantor-spectrum statement is known there); the treatment "does not cover the Liouville frequencies", since the reducibility input fails for them, and the non-critical DTMP beyond Diophantine frequencies is described as "notoriously hard"; the authors nevertheless expect the DTMP to hold for all frequencies, pointing to the Avila–You–Zhou strategy as a possible attack.

Problem Statement

Determine whether the Dry Ten Martini property holds for the unitary almost Mathieu operator Wλ1,λ2,Φ,θ=Sλ1Qλ2,Φ,θW_{\lambda_1,\lambda_2,\Phi,\theta}=S_{\lambda_1}Q_{\lambda_2,\Phi,\theta} in the regimes left open by Cedzich–Li (2025), thereby completing the Dry Ten Martini Problem for the UAMO (the non-critical regime λ1≠λ2\lambda_1\neq\lambda_2 with Diophantine Φ\Phi being already settled):

(i) Critical regime: for every 0<λ1=λ2≤10<\lambda_1=\lambda_2\leq1 and every irrational frequency Φ\Phi (Diophantine or Liouville), is every gap of the spectrum Σλ1,λ2,Φ\Sigma_{\lambda_1,\lambda_2,\Phi} allowed by the Gap Labeling Theorem — every label kΦ/2 mod 1k\Phi/2 \bmod 1, k∈Zk\in\mathbb{Z} — open rather than collapsed?

(ii) Non-critical regime with non-Diophantine frequency: for every 0<λ1≠λ2≤10<\lambda_1\neq\lambda_2\leq1 and every irrational Φ\Phi outside the Diophantine class (Liouville frequencies), is every allowed gap open?

Equivalently: prove, or disprove, that for every irrational Φ\Phi and every coupling pair (λ1,λ2)∈(0,1]2(\lambda_1,\lambda_2)\in(0,1]^2, all gaps labeled by kΦ/2 mod 1k\Phi/2 \bmod 1, k∈Zk\in\mathbb{Z}, of Σλ1,λ2,Φ\Sigma_{\lambda_1,\lambda_2,\Phi} are open. A disproof consists of explicit parameters (λ1,λ2,Φ)(\lambda_1,\lambda_2,\Phi) in the open regimes and an integer kk such that the gap with label kΦ/2 mod 1k\Phi/2 \bmod 1 is collapsed (the label is realized without a nondegenerate open gap), together with a rigorous proof of the collapse.

References

  1. Christopher Cedzich and Long Li. Twenty dry Martinis for the unitary almost-Mathieu operator. Ergodic Theory and Dynamical Systems 46:903-920, 2025 (published online 10 December 2025; print issue April 2026). DOI: 10.1017/etds.2025.10259; arXiv:2503.06710. https://arxiv.org/abs/2503.06710 (source paper)
  2. Artur Avila and Svetlana Jitomirskaya. The Ten Martini Problem. Annals of Mathematics 170(1):303-342, 2009. DOI: 10.4007/annals.2009.170.303. https://doi.org/10.4007/annals.2009.170.303
  3. Artur Avila, Jiangong You, and Qi Zhou. Dry Ten Martini Problem in the non-critical case. arXiv:2306.16254, 2023 (revised 2024). https://arxiv.org/abs/2306.16254
  4. Christopher Cedzich, Jake Fillman, and Darren C. Ong. Almost Everything About the Unitary Almost-Mathieu Operator. Communications in Mathematical Physics 403:745-794, 2023. DOI: 10.1007/s00220-023-04808-4; arXiv:2112.03216. https://arxiv.org/abs/2112.03216
  5. Man-Duen Choi, George A. Elliott, and Noriko Yui. Gauss polynomials and the rotation algebra. Inventiones Mathematicae 99(1):225-246, 1990. DOI: 10.1007/BF01234419. https://doi.org/10.1007/BF01234419
  6. Joaquim Puig. Cantor Spectrum for the Almost Mathieu Operator. Communications in Mathematical Physics 244(2):297-309, 2004. DOI: 10.1007/s00220-003-0977-3. https://doi.org/10.1007/s00220-003-0977-3
  7. Barry Simon. Almost periodic Schrödinger operators: A Review. Advances in Applied Mathematics 3(4):463-490, 1982. DOI: 10.1016/s0196-8858(82)80018-3. https://doi.org/10.1016/s0196-8858(82)80018-3
  8. Jordi-Lluís Figueras and Joaquim Puig. Computer Validation of Open Gaps for the Almost Mathieu Operator with Critical Coupling. arXiv:2410.18536, 2024 (revised 2025). https://arxiv.org/abs/2410.18536
  9. Fan Yang. Arithmetic spectral transition for the unitary almost Mathieu operator. arXiv:2512.24616, 2025. https://arxiv.org/abs/2512.24616
  10. Xinyu Yang, Long Li, and Qi Zhou. Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks. arXiv:2601.20081, 2026. https://arxiv.org/abs/2601.20081
  11. Xianzhe Li, Disheng Xu, and Qi Zhou. Monotonicity, global symplectification and the stability of Dry Ten Martini Problem. arXiv:2601.02222, 2026 (revised April 2026). https://arxiv.org/abs/2601.02222
  12. Lingrui Ge, Yiqian Wang, and Jiahao Xu. The Dry Ten Martini Problem for C2C^2 cosine-type quasiperiodic Schrödinger operators. arXiv:2503.06918, 2025. https://arxiv.org/abs/2503.06918
  13. Ram Band, Siegfried Beckus, and Raphael Loewy. The Dry Ten Martini Problem for Sturmian Hamiltonians. arXiv:2402.16703, 2024. https://arxiv.org/abs/2402.16703
  14. Rui Han. Dry ten martini problem for the non-self-dual extended Harper's model. arXiv:1607.08571, 2016. https://arxiv.org/abs/1607.08571
  15. Jiawei He, Yuan Shan, and Yongjian Wang. Cantor spectrum via a reducibility-duality bridge for the mosaic almost Mathieu operator. SSRN preprint, 2026. DOI: 10.2139/ssrn.7083575. https://doi.org/10.2139/ssrn.7083575
  16. Quan Lin, Christopher Cedzich, Qi Zhou, and Peng Xue. Observation of Metal-Insulator and Spectral Phase Transitions in Aubry-André-Harper Models. arXiv:2508.08255, 2025. https://arxiv.org/abs/2508.08255