Dry Ten Martini Problem for type-I energies of quasiperiodic Schrödinger operators with arbitrary analytic potentials

For a one-frequency quasiperiodic Schrödinger operator with irrational frequency α and real-analytic potential v, the gap-labelling theorem assigns to each spectral gap an integer label: the integrated density of states is constant on the gap with value kα mod 1. The Dry Ten Martini Problem asks whether all labelled gaps are open. For general analytic potentials gaps can collapse, so Ge–Jitomirskaya–You conjectured that for every irrational α and every analytic v, every type-I energy (an energy whose T-acceleration equals 1, i.e. the complexified Lyapunov exponent undergoes exactly one transition) carrying a gap label is the boundary of an open gap; Li–Xu–Zhou's Conjecture 1.2, "All gaps of Type I operators are open", is the related operator-level variant, restricted to potentials whose every spectral energy is type-I, and follows from the per-energy conjecture. The conjecture is proved in the supercritical regime for analytic Type I potentials at every irrational frequency (Li–Xu–Zhou, 2026), and in the subcritical regime for trigonometric-polynomial potentials (Ge–Jitomirskaya–You for β(α)=0, extendable to all irrational α via Avila–You–Zhou), but remains open in the critical regime — including the critical almost-Mathieu operator itself — and open for the per-energy statement for arbitrary analytic potentials: the published theorems either assume the whole operator is Type I or restrict to trigonometric polynomials, while unpublished Argentieri–Avila counterexamples produce interval spectrum arbitrarily close to the subcritical almost-Mathieu operator. The problem is to prove or refute the conjecture in the full analytic class.

Background

A one-frequency quasiperiodic Schrödinger operator on $\ell^{2}(\mathbb{Z})$ acts as $(H_{v,\alpha,x}u)_n=u_{n+1}+u_{n-1}+v(x+n\alpha)u_n,$ where $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ is the frequency, $x\in\mathbb{T}=\mathbb{R}/\mathbb{Z}$ is the phase, and $v:\mathbb{T}\to\mathbb{R}$ is a real-analytic potential. The almost-Mathieu operator (AMO), the special case $v(x)=2\lambda\cos(2\pi x)$ , is the standard tight-binding model of an electron on a lattice in a magnetic field and underlies the Hofstadter butterfly. The spectrum $\Sigma_{v,\alpha}$ is a compact set independent of $x$ ; the bounded components of its complement are the spectral gaps. The integrated density of states (IDS) $N_{v,\alpha}(E)$ , obtained as the thermodynamic limit of eigenvalue counting functions, is continuous, strictly increasing on $\Sigma_{v,\alpha}$ , and constant on each gap. The gap-labelling theorem (Johnson–Moser) assigns a topological label to each bounded gap: on a gap, $N_{v,\alpha}\equiv k\alpha\pmod{\mathbb{Z}}$ for a unique nonzero integer $k$ ; in the quantum-Hall interpretation of these models the labels are Chern numbers of quantized Hall conductances.

The Ten Martini Problem asks whether the spectrum is a Cantor set; the stronger Dry Ten Martini Problem (DTMP) asks whether every gap allowed by the gap-labelling theorem is actually open, i.e. no labelled gap collapses to a point. For the AMO, the Ten Martini Problem was resolved by Avila–Jitomirskaya, and the dry version is known in the non-critical case $\lambda\neq\pm1$ (Avila–You–Zhou); the critical case $\lambda=\pm1$ remains open and is itself a special case of the conjecture below, since every AMO spectral energy is type-I at any coupling. For general analytic potentials the DTMP is false as stated — spectra may contain intervals, so labelled gaps can collapse — and the question becomes to identify the class of energies at which openness can be guaranteed.

Avila's global theory of one-frequency Schrödinger operators supplies the dynamical framework. To the eigenvalue equation one associates the Schrödinger cocycle $(\alpha,S_E^{v})$ , where $S_E^{v}(x)$ is the $2\times2$ matrix $\begin{pmatrix}E-v(x)&-1\\1&0\end{pmatrix}$ , and studies the Lyapunov exponent $L_\varepsilon(E)$ of the complexified cocycle with the base phase shifted by $i\varepsilon$ : a convex, piecewise-affine function of $\varepsilon\ge0$ . An energy is supercritical if $L_\varepsilon(E)>0$ for all small $\varepsilon>0$ , subcritical if $L_\varepsilon(E)\equiv0$ near $0$ , and critical otherwise; the number of affine pieces traversed is the acceleration $\omega(E)$ . Ge–Jitomirskaya–You introduced the T-acceleration $\bar{\omega}(E)=\lim_{\varepsilon\to\varepsilon_1^{+}}\frac{L_{\varepsilon}(E)-L_{\varepsilon_1}(E)}{2\pi(\varepsilon-\varepsilon_1)},$ where $\varepsilon_1$ is the first turning point of $\varepsilon\mapsto L_\varepsilon(E)$ , with $\bar{\omega}(E)=1$ if no turning point exists, and they call $E$ a type-I energy when $\bar{\omega}(E)=1$ (this is the convention of Ge–Jitomirskaya–You, the originators of the notion; Li–Xu–Zhou instead set $\bar{\omega}(E)=0$ when no turning point exists, which changes the type-I class only for energies without a turning point). For trigonometric-polynomial potentials this is equivalent to the Aubry-dual cocycle — the transfer cocycle of the Fourier-dual finite-band operator produced by Aubry duality — being partially hyperbolic with a two-dimensional center bundle, that is, split into stable and unstable directions together with exactly one neutral two-dimensional block. Type-I energies form a class stable in the analytic topology that interpolates between the sub- and supercritical regimes; for the AMO every energy in the spectrum is type-I, so the general statement below reduces to the classical DTMP in that case. The arithmetic of the frequency enters through $\beta(\alpha)=\limsup_n\log q_{n+1}/q_n$ , with $q_n$ the denominators of the continued-fraction convergents: $\beta(\alpha)=0$ is the Diophantine case, while $\beta(\alpha)>0$ (the Liouvillean case) is where the gaps are smallest and most irregularly distributed.

Ge–Jitomirskaya–You proved Cantor spectrum for type-I trigonometric-polynomial operators and conjectured the corresponding dry statement; Li–Xu–Zhou formulated the operator-level Conjecture 1.2 (all gaps of Type I operators — those whose every spectral energy is type-I — are open) and proved its supercritical case for every analytic Type I potential at every irrational frequency via a new geometric, all-frequency method: a fibred rotation number for the projective action on the Lagrangian Grassmannian (the manifold of Lagrangian subspaces on which symplectic cocycles act naturally), monotonicity of one-parameter families of Hermitian-symplectic cocycles (cocycles preserving both a symplectic and a Hermitian form), and a global symplectification — a holonomy-driven parallel transport reducing the dynamics to a two-dimensional center. Their proof first handles trigonometric-polynomial potentials, for which the Aubry dual is a finite-band operator with a finite-dimensional cocycle, and then passes to general analytic Type I potentials by controlling the dependence of the estimates on the truncation dimension, through a dimension-free Aubry duality in weighted analytic norms and the convergence of the two-dimensional intrinsic center cocycle. The earlier subcritical proof for trigonometric-polynomial potentials (Ge–Jitomirskaya–You for $\beta(\alpha)=0$ , extended to all irrational $\alpha$ through Avila–You–Zhou's quantitative Aubry duality) relies on the finite Fourier support of $v$ . Whether the per-energy conjecture holds for arbitrary analytic $v$ remains open — Li–Xu–Zhou's Theorem 1.3 assumes the whole potential is Type I and covers only the supercritical regime — and they note that unpublished counterexamples of Argentieri–Avila produce interval spectrum arbitrarily close to the subcritical AMO, so any argument must use the type-I hypothesis in an essential way.

Problem Statement

Prove or refute the following conjecture (Conjecture 1 of Ge–Jitomirskaya–You; Conjecture 1.2 of Li–Xu–Zhou, "All gaps of Type I operators are open", is the operator-level variant that follows from it on the class of potentials whose every spectral energy is type-I). For every irrational frequency $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and every real-analytic potential $v\in C^{\omega}(\mathbb{T},\mathbb{R})$ , each type-I energy $E_k\in\Sigma_{v,\alpha}$ (that is, $\bar{\omega}(E_k)=1$ ) satisfying the gap-labelling condition $N_{v,\alpha}(E_k)\equiv k\alpha\pmod{\mathbb{Z}}$ for some $k\in\mathbb{Z}\setminus\{0\}$ is a boundary of an open spectral gap of $H_{v,\alpha,x}$ .

The open content is the following: the critical dynamical regime, open even for trigonometric polynomials — at the critical almost-Mathieu operator ( $\lambda=\pm1$ ) every spectral energy is type-I, so the conjecture contains the last open case of the classical Dry Ten Martini Problem; the subcritical regime beyond trigonometric-polynomial potentials, which Li–Xu–Zhou note may extend to all irrational frequencies and general analytic Type I potentials but leave unproved; and the per-energy statement for arbitrary analytic potentials that are not Type I operators — the published supercritical theorem assumes every spectral energy of the operator is type-I, whereas the conjecture requires only that the individual labelled energy be type-I, for potentials with finitely or infinitely many nonzero Fourier coefficients. In the supercritical regime for analytic Type I potentials, and in the subcritical regime for trigonometric-polynomial potentials, the conjecture is already a theorem. A full resolution is either a proof valid for arbitrary analytic $v$ , or an explicit real-analytic potential $v$ , an irrational frequency $\alpha$ , and a type-I energy carrying a gap label that is not the boundary of any open gap.

References

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