The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators at arbitrary irrational frequencies

Ge, Wang, and Xu (2025) proved the Dry Ten Martini Problem for the discrete quasiperiodic Schrödinger operator $H_{\alpha,\lambda v,x}$ on $\ell^2(\mathbb{Z})$ when the potential $v$ is $C^2$ and cosine-type, the coupling $\lambda$ is sufficiently large, and the frequency $\alpha$ is Diophantine, and stated that they expect the result for all irrational $\alpha$. The open problem is the arithmetic-free extension: prove, or disprove, that for every irrational $\alpha$ (including Liouville frequencies outside every Diophantine class) and every $C^2$ cosine-type $v$ there is a finite $\lambda_0(\alpha,v)$ such that for all $\lambda\ge\lambda_0$ every spectral gap allowed by the gap-labelling theorem is open.

Background

A discrete one-frequency quasiperiodic Schrödinger operator acts on sequences $u\in\ell^2(\mathbb{Z})$ by $(H_{\alpha,\lambda v,x}u)_n=u_{n+1}+u_{n-1}+\lambda v(x+n\alpha)u_n,$ where the sampling function (the potential) $v:\mathbb{T}\to\mathbb{R}$ is a real function on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, $\alpha$ is an irrational frequency, $\lambda\in\mathbb{R}$ is the coupling constant, and $x\in\mathbb{T}$ is the phase. For irrational $\alpha$ the spectrum $\Sigma_{\alpha,\lambda v}\subset\mathbb{R}$ is a compact set independent of the phase; the bounded connected components of its complement are the spectral gaps. The integrated density of states (IDS) $N(E)$ — the phase-averaged counting function of eigenvalues per unit volume — is constant on each open gap, and the gap-labelling theorem of Johnson and Moser associates to each open gap an integer label $k$ through $N(E)=k\alpha\bmod 1$. The operator has a dry Cantor spectrum when every label $k\in\mathbb{Z}$ allowed by this theorem is realized by a nonempty open gap, i.e. the spectrum is a Cantor set with all gaps open.

The question originates with a 1981 problem for the almost Mathieu operator (the case $v(x)=2\cos 2\pi x$), famously attached to a reward of ten martinis and popularized as the Ten Martini Problem asking whether the spectrum is a Cantor set, as recounted in Simon's 1982 review; the original all-gaps question became known as the Dry Ten Martini Problem (DTM). For the almost Mathieu operator the Ten Martini Problem was completely solved by Avila and Jitomirskaya (Cantor spectrum for all irrational $\alpha$ and nonzero coupling), and the DTM was solved in the non-critical regime $|\lambda|\neq 1$ for every irrational $\alpha$ by Avila, You, and Zhou. These proofs use analyticity of the potential in an essential way, through Aubry duality (a Fourier-transform symmetry relating the operator to a dual operator with inverted coupling) and quantitative almost-reducibility of the associated $\mathrm{SL}(2,\mathbb{R})$ Schrödinger cocycles — the transfer matrices $A^{(E)}(x)$ encoding the recursion $u_{n+1}=(\lambda v(x+n\alpha)-E)u_n-u_{n-1}$, iterated along the orbit $x+n\alpha$, whose asymptotic exponential growth rate is the Lyapunov exponent.

For non-analytic potentials much less is known. The model class studied here goes back to Sinai (1987). A potential $v\in C^2(\mathbb{T},\mathbb{R})$ is called cosine-type if $dv/dx$ vanishes at exactly two points of $\mathbb{T}$, one the global minimum and the other the global maximum, both non-degenerate in the sense that $d^2v/dx^2\neq 0$ there; this is the smooth analogue of the shape of $2\cos 2\pi x$. Wang and Zhang proved that for such potentials, Diophantine frequencies, and large coupling the spectrum is a Cantor set (gaps dense), and Ge, Wang, and Xu completed the dry version. A frequency $\alpha$ is Diophantine, written $\alpha\in DC_{\tau,\gamma}$, if $\inf_{j\in\mathbb{Z}}|\alpha-p/q-j|\geq\gamma/|q|^{\tau}$ for all integers $p,q$ with $q\neq 0$; the union $DC$ over all $\tau>2$, $\gamma>0$ has full Lebesgue measure, but its complement — which contains the Liouville frequencies, those approximable by rationals faster than any power — is a dense set of measure zero. Ge, Wang, and Xu proved that if $\alpha\in DC$ and $v$ is $C^2$ cosine-type, there exists $\lambda_0(\alpha,v)$ such that for all $\lambda\ge\lambda_0$: the DTM holds for $H_{\alpha,\lambda v,x}$ for every phase $x$ (Theorem 1); the spectrum is homogeneous, meaning there is $\mu>0$ with $|\Sigma_{\alpha,\lambda v}\cap(E-\epsilon,E+\epsilon)|>\mu\epsilon$ whenever $\epsilon<\mathrm{diam}\,\Sigma_{\alpha,\lambda v}$ and $E\in\Sigma_{\alpha,\lambda v}$ (Theorem 2; this no-large-holes property is used in inverse spectral theory); and the IDS is absolutely continuous (Theorem 3). In addition, each gap $G_k^\lambda$ with label $k$ obeys the two-sided size bound $c\lambda^{-10000|k|}\leq|G_k^\lambda|\leq C\lambda^{-|k|/10000}$ (Theorem 8), with visibly non-optimal exponents. The authors stress that the cosine-type geometry, unlike the frequency arithmetic, is essential: outside the cosine-type class the dry conclusion might fail even for analytic potentials, since the acceleration of the Lyapunov exponent — an integer-valued invariant in the global theory of one-frequency Schrödinger cocycles — equals one for cosine-type potentials but may be larger otherwise. Their Remark 1 states that the Diophantine condition on $\alpha$ "is not essential" and that they "expect the above result holds for all irrational $\alpha$".

Problem Statement

Let $v\in C^2(\mathbb{T},\mathbb{R})$ be a cosine-type potential, i.e. $v'$ vanishes at exactly two points of the circle, the non-degenerate global minimum and the non-degenerate global maximum, and let $\alpha$ be an arbitrary irrational frequency, with no Diophantine or other arithmetic hypothesis.

Dry Ten Martini at arbitrary irrational frequency (core). Prove, or disprove: there exists $\lambda_0(\alpha,v)<\infty$ such that for every $\lambda\ge\lambda_0$ and every phase $x\in\mathbb{T}$, every spectral gap of $H_{\alpha,\lambda v,x}$ allowed by the gap-labelling theorem is open — equivalently, the spectrum $\Sigma_{\alpha,\lambda v}$ is a dry Cantor set whose open gaps realize all labels $k\in\mathbb{Z}$. Theorem 1 of Ge–Wang–Xu establishes this for $\alpha\in DC$; the open case is the arithmetic complement, in particular Liouville frequencies.

A refutation fixes an irrational non-Diophantine $\alpha$ and a cosine-type $v$, and gives a sequence of couplings $\lambda_n\to\infty$ together with, for each $n$, a label $k_n\in\mathbb{Z}$ allowed by the gap-labelling theorem and a rigorous certificate that no open gap of $H_{\alpha,\lambda_n v,x}$ carries label $k_n$.

References

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