The Dry Ten Martini Problem for the extended Harper's model: the self-dual region III and Liouvillean frequencies

The extended Harper's model (EHM) is the quasiperiodic Jacobi operator modeling an electron on a two-dimensional lattice in a perpendicular magnetic field when both nearest-neighbor and next-nearest-neighbor hopping are allowed; its three positive couplings split the parameter space into two dual non-self-dual regions (I and II) and a self-dual region III. Han (2018) proved the dry Ten Martini problem for the EHM — Cantor spectrum with all gap-labelling-predicted spectral gaps open — in regions I and II for Diophantine frequencies. The problem stated here is the remaining part: decide whether every label $\ell\in\alpha\mathbb Z+\mathbb Z\cap(0,1)$ of the gap-labelling theorem is realized by an open gap (a) in the self-dual region III, $\max(1,\lambda_2)<\lambda_1+\lambda_3$, for irrational frequency $\alpha$, where even Cantor spectrum is known only on the isotropic slice $\lambda_1=\lambda_3$, and (b) in regions I and II for Liouvillean (non-Diophantine) frequencies $\beta(\alpha)>0$, where even Cantor spectrum is open. Both regimes remain open.

Background

A quasiperiodic operator is a family of self-adjoint operators on $\ell^2(\mathbb Z)$ obtained by shifting the coefficients of a periodic sequence along the orbit $\theta+n\alpha$ of an irrational rotation; physically it describes a particle on a one-dimensional lattice whose couplings are modulated incommensurately with the lattice, the standard reduction of an electron on a two-dimensional lattice in a perpendicular magnetic field (the magnetic flux through a unit cell is the frequency $\alpha$). The prototype is the almost Mathieu operator (AMO), $(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda\cos(2\pi(\theta+n\alpha))u_n$ with phase $\theta\in\mathbb T=\mathbb R/\mathbb Z$. The Ten Martini Problem, named after a wager of Mark Kac relayed by Barry Simon, asked whether the AMO spectrum is a Cantor set (a closed, nowhere dense, perfect set) for every irrational $\alpha$; Avila and Jitomirskaya (2009) proved that it is, for all irrational $\alpha$ and $\lambda\neq0$.

The finer, 'dry' version asks whether all spectral gaps predicted by the gap-labelling theorem are actually open. The integrated density of states (IDS) $N(E)$ is the distribution function of the spectral ensemble averaged over the phase $\theta$; the gap-labelling theorem states that whenever $E$ lies at the edge of a spectral gap, $N(E)$ belongs to the subgroup $\alpha\mathbb Z+\mathbb Z$ (values in $[0,1]$). 'Dry Ten Martini' means every label $\ell\in\alpha\mathbb Z+\mathbb Z\cap(0,1)$ is attained at the edge of a nonempty (open) gap. For the AMO this was proved for Liouvillean frequencies by Choi, Elliott and Yui via Gauss polynomials ($q$-binomial coefficient sums) and the rotation algebra (the $C^*$-algebra generated by the position and magnetic-translation unitaries, in which the gap labels arise as traces of spectral projections), for a positive-measure set of parameters by Puig, for Diophantine frequencies with $\lambda\neq1$ by Avila and Jitomirskaya using almost localization (a quantitative substitute for Anderson localization, the spatial-decay mechanism for eigenfunctions) and its dual notion of almost reducibility (approximate conjugacy of the associated transfer-matrix cocycles to constant ones), and for all irrational frequencies in the non-critical cases $\lambda\neq\pm1$ by Avila, You and Zhou (2023). The critical case $\lambda=1$ has been approached only by physics-style arguments.

The extended Harper's model, proposed by Thouless in 1983, generalizes the AMO by allowing hops to next-nearest-neighbor sites, which after gauge reduction produces a Jacobi operator (a doubly infinite tri-diagonal self-adjoint matrix) with nonconstant off-diagonal coefficients: $(H_{\lambda_1,\lambda_2,\lambda_3,\alpha,\theta}u)_n=c(\theta+n\alpha)\,u_{n+1}+\tilde c(\theta+(n-1)\alpha)\,u_{n-1}+2\cos(2\pi(\theta+n\alpha))\,u_n,$ where $c(\theta)=\lambda_1 e^{-2\pi i(\theta+\alpha/2)}+\lambda_2+\lambda_3 e^{2\pi i(\theta+\alpha/2)}$ and $\tilde c(\theta)=\overline{c(\theta)}$; $\lambda_2$ weights the nearest-neighbor hopping and $\lambda_1,\lambda_3$ the two next-nearest-neighbor hoppings. The model retains an Aubry–André duality, a Fourier-type map $\sigma(\lambda_1,\lambda_2,\lambda_3)=(\lambda_3/\lambda_2,1/\lambda_2,\lambda_1/\lambda_2)$ sending the operator to a dual operator of the same family. The parameter space of positive couplings splits into three regions: region I, $0<\max(\lambda_1+\lambda_3,\lambda_2)<1$, where the Lyapunov exponent (the asymptotic exponential growth rate of the transfer matrices along the orbit) is positive (supercritical); region II, $0<\max(\lambda_1+\lambda_3,1)<\lambda_2$ (subcritical); and region III, $0<\max(1,\lambda_2)<\lambda_1+\lambda_3$, which is mapped to itself by $\sigma$ and is therefore called self-dual. Regions I and II are interchanged by the duality, exactly as the supercritical and subcritical halves of the AMO line; region III is the EHM analogue of the critical AMO.

Three bodies of prior knowledge frame the open question. First, Jitomirskaya and Marx (2012) computed the exact Lyapunov exponent of the EHM for all couplings, developing a theory of analytic cocycles with singularities (the coefficient $c(\theta)$ can vanish, so the usual matrix-cocycle framework degenerates). Second, Avila, Jitomirskaya and Marx (2017) completed the description of the spectral measures — the phase-averaged decomposition into absolutely continuous parts (spectral measures absolutely continuous with respect to Lebesgue measure, giving extended states) and singular continuous parts — over the whole self-dual region III for every irrational $\alpha$, discovering a 'spectral collapse' from purely absolutely continuous to purely singular continuous spectrum when the next-nearest-neighbor couplings become symmetric ($\lambda_1=\lambda_3$); for the spectrum as a set, Cantor spectrum in region III is established only in that isotropic case. Third, Han (2018) proved the dry Ten Martini property in regions I and II for Diophantine frequencies — those with $\beta(\alpha)=0$, where $\beta(\alpha)=\limsup_n \ln q_{n+1}/q_n$ is built from the denominators $q_n$ of the continued-fraction convergents of $\alpha$; frequencies with $\beta(\alpha)>0$ are called Liouvillean and have extraordinarily good rational approximations. Han's proof, following Avila–Jitomirskaya, establishes almost localization in region I and converts it by quantitative Aubry duality into almost reducibility in region II, then opens each collapsed gap by a perturbative argument. The two regimes left open by that theorem are precisely the ones in which this strategy breaks down: region III is invariant under the duality, so no better-behaved dual partner exists, and for Liouvillean frequencies the quantitative localization and duality estimates are not uniform in $\beta(\alpha)$.

Problem Statement

Let $H_{\lambda_1,\lambda_2,\lambda_3,\alpha,\theta}$ be the extended Harper's model on $\ell^2(\mathbb Z)$ with positive couplings $\lambda_1,\lambda_2,\lambda_3>0$, irrational frequency $\alpha\in\mathbb R\setminus\mathbb Q$, and phase $\theta\in\mathbb T=\mathbb R/\mathbb Z$: $(H_{\lambda_1,\lambda_2,\lambda_3,\alpha,\theta}u)_n=c(\theta+n\alpha)\,u_{n+1}+\tilde c(\theta+(n-1)\alpha)\,u_{n-1}+2\cos(2\pi(\theta+n\alpha))\,u_n,$ where $c(\theta)=\lambda_1 e^{-2\pi i(\theta+\alpha/2)}+\lambda_2+\lambda_3 e^{2\pi i(\theta+\alpha/2)}$ and $\tilde c(\theta)=\overline{c(\theta)}$. Its spectrum $\Sigma_{\lambda_1,\lambda_2,\lambda_3,\alpha}\subset\mathbb R$, which does not depend on the phase $\theta$, is compact; a spectral gap is a bounded connected component of the complement $\mathbb R\setminus\Sigma_{\lambda_1,\lambda_2,\lambda_3,\alpha}$, and a gap is open if it is nonempty. Let $N(E)$ denote the integrated density of states, the distribution function of the spectral measure averaged uniformly over the phase $\theta$. By the gap-labelling theorem, $N$ is constant on each spectral gap and takes values in the subgroup $\alpha\mathbb Z+\mathbb Z$ there; the dry Ten Martini property is the assertion that $\Sigma_{\lambda_1,\lambda_2,\lambda_3,\alpha}$ is a Cantor set and that every label $\ell\in\alpha\mathbb Z+\mathbb Z\cap(0,1)$ is the constant value of $N$ on some open gap.

Write $p_n/q_n$ for the convergents of the continued-fraction expansion of $\alpha$ and $\beta(\alpha)=\limsup_{n\to\infty}\frac{\ln q_{n+1}}{q_n}$; $\alpha$ is called Diophantine if $\beta(\alpha)=0$ and Liouvillean if $\beta(\alpha)>0$. Han (2018) proved the dry Ten Martini property for Diophantine $\alpha$ in the non-self-dual region I ($0<\max(\lambda_1+\lambda_3,\lambda_2)<1$) and region II ($0<\max(\lambda_1+\lambda_3,1)<\lambda_2$). The problem stated here — the regimes not settled by that theorem — is to decide:

(a) in the self-dual region III, $0<\max(1,\lambda_2)<\lambda_1+\lambda_3$, for irrational $\alpha$ — where even Cantor spectrum is currently known only in the isotropic case $\lambda_1=\lambda_3$; and

(b) in region I ($0<\max(\lambda_1+\lambda_3,\lambda_2)<1$) and region II ($0<\max(\lambda_1+\lambda_3,1)<\lambda_2$), for Liouvillean frequencies $\beta(\alpha)>0$ — where even Cantor spectrum is open:

whether the spectrum is a Cantor set with all gaps open, i.e. whether every label $\ell\in\alpha\mathbb Z+\mathbb Z\cap(0,1)$ is attained as $N(E)$ at the edge of a nonempty spectral gap. A rigorous proof of this property in either regime, or a rigorously established counterexample — explicit couplings, an irrational frequency, and an admissible label realized by no open gap in that regime — resolves the corresponding part of the problem. No restriction on couplings, phases, energy ranges, or method is imposed beyond the regimes stated.

References

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