In a 2008 preprint, A. Avila proved that the spectral measures of the almost Mathieu operator are absolutely continuous if and only if the coupling satisfies ∣λ∣<1|\lambda|<1, for every irrational frequency and every phase, and formulated a 'subcritical program': extend such every-phase absolute-continuity results to general one-frequency analytic quasiperiodic Schrödinger operators, without arithmetic restrictions on the frequency. One component of the source's open list — the conjecture that the critical (∣λ∣=1|\lambda|=1) almost Mathieu operator has purely singular continuous spectrum for every irrational frequency and every phase — was resolved by Jitomirskaya (Adv. Math. 392 (2021) 107997). The dynamical core of the program, the Almost Reducibility Conjecture, is settled in Avila's global-theory framework (the general-analytic case being documented in his announced and partly preprinted manuscripts), and published alternative proofs cover trigonometric polynomial potentials; almost-every-phase spectral consequences are in print. The surviving open core is (i) to prove, for general real-analytic potentials in the small-coupling almost-localization regime and exponentially Liouville frequencies (β(α)>0\beta(\alpha)>0), that all spectral measures are absolutely continuous for every phase — the extension of the source's Theorem 3.2 beyond the subexponential regime β(α)=0\beta(\alpha)=0. A further structural point that the source itself recorded as unresolved, and that the subsequent literature has left open, is (ii) whether an energy can correspond to uncountably many dual phases on an exceptional energy set. A 2013 preprint of Liu and Yuan claims (i) under additional hypotheses but was never published and has no recorded citations; no peer-reviewed proof exists, so the core remains open.

Background

A quasiperiodic Schrödinger operator on the Hilbert space ℓ2(Z)\ell^2(\mathbb{Z}) of square-summable sequences is the discrete Jacobi-type operator

(Hλv,α,θu)n=un+1+un−1+λv(θ+nα)un,(H_{\lambda v,\alpha,\theta}u)_n = u_{n+1} + u_{n-1} + \lambda v(\theta + n\alpha)u_n,

where v:R/Z→Rv:\mathbb{R}/\mathbb{Z}\to\mathbb{R} is a real-analytic 1-periodic potential, α\alpha is an irrational frequency, θ\theta is a phase, and λ\lambda is a coupling constant. The almost Mathieu operator (AMO) is the special case v(θ)=2cos⁡(2πθ)v(\theta)=2\cos(2\pi\theta); it models an electron on a two-dimensional square lattice in a perpendicular magnetic field of flux α\alpha (the Harper/Azbel–Hofstadter model). By the spectral theorem, the spectral measures of such operators decompose into pure point, singular continuous, and absolutely continuous parts, and a central question of the field is which pure type occurs for which parameters.

The arithmetic strength of the frequency is measured by β(α)=lim sup⁡n→∞ln⁡qn+1/qn\beta(\alpha)=\limsup_{n\to\infty}\ln q_{n+1}/q_n, where qnq_n are the denominators of the continued-fraction convergents to α\alpha. Diophantine frequencies (badly approximated by rationals) satisfy β=0\beta=0; frequencies with β>0\beta>0, called exponentially Liouville, are so well approximated by rationals that perturbative (KAM-type) methods break down. The strength of phase resonances is measured analogously by δ(α,θ)=lim sup⁡∣k∣→∞−ln⁡∥2θ+kα∥R/Z∣k∣\delta(\alpha,\theta)=\limsup_{|k|\to\infty}\frac{-\ln\|2\theta+k\alpha\|_{\mathbb{R}/\mathbb{Z}}}{|k|}, the exponential rate at which the rotated phase approaches a resonance, that is, a point where 2θ+kα2\theta+k\alpha is an integer.

Aubry duality is a Fourier-transform correspondence between the family {Hλv,α,θ}θ\{H_{\lambda v,\alpha,\theta}\}_{\theta} and a dual family of operators H^\hat H; for the AMO it exchanges coupling with inverse coupling and localized with extended states. In the quantitative form developed by Avila and Jitomirskaya (J. Eur. Math. Soc. 12 (2010)), each energy EE of the primal operator is associated with one or more dual phases θ(E)\theta(E) through localized eigenfunctions of the dual model. The source paper calls the potential vv small when the dual family {H^v,α,θ}θ\{\hat H_{v,\alpha,\theta}\}_{\theta} is almost localized — a quantitative localization property introduced by Avila and Jitomirskaya, meaning that for almost every dual phase the dual operators have exponentially localized eigenfunctions with bounds uniform outside a small, explicitly controlled resonant set of phases; v=2λcos⁡v=2\lambda\cos is small exactly when 0<λ<10<\lambda<1, and by Theorem 5.1 of Avila–Jitomirskaya every sufficiently small multiple λv\lambda v is small at least in the subexponential regime β=0\beta=0.

The dynamical reformulation treats the eigenvalue equation as a matrix cocycle over the circle rotation x↦x+αx\mapsto x+\alpha. Avila's global theory (Acta Math. 215 (2015) 1–54) trichotomizes analytic one-frequency Schrödinger cocycles into subcritical, critical, and supercritical regimes via the complexified Lyapunov exponent (the exponential growth rate of transfer-matrix products, analytically continued in the phase) and its acceleration (an integer invariant that measures how the exponent changes under analytic continuation of the phase, separating the three regimes). The Almost Reducibility Conjecture (ARC) asserts that every subcritical cocycle is almost reducible (a limit of cocycles analytically conjugate to constants); it is the dynamical core of the subcritical program.

The source paper (Avila, arXiv:0810.2965) proved the Main Theorem: the spectral measures of the AMO are absolutely continuous if and only if ∣λ∣<1|\lambda|<1, for every irrational α\alpha and every θ\theta — the subcritical half, in full generality, of a physics conjecture of Aubry and André (whose measure-theoretic version was proved by Jitomirskaya in 1999). Its Section 3.2 generalizes the subexponential-regime argument: Theorem 3.2 states that if vv is small and β(α)=0\beta(\alpha)=0, then the spectral measures of Hv,α,θH_{v,\alpha,\theta} are absolutely continuous. Its Section 1.3 formulates the subcritical program for general analytic potentials — one goal being pure absolutely continuous spectrum 'in full generality' — and identifies the exponential regime β>0\beta>0 as the part where the paper's estimates are specific to the AMO, while noting that the paper's cancellation technique 'is the only tool available for deducing pure absolutely continuous spectrum for every phase in the case of very Liouvillean frequencies' and is expected to extend once AMO-specific inputs are replaced. A footnote of the source records an unresolved structural question of this duality: each energy usually corresponds to finitely many dual phases, but the source could not rule out that on an exceptional energy set (of Hausdorff dimension zero in its analysis) there could be uncountably many, an issue tied to the possible coexistence of point and singular continuous spectrum in the dual model.

Remark 1.1 of the source also restated the critical-coupling conjecture (explicit in Jitomirskaya's 1999 Annals paper): for ∣λ∣=1|\lambda|=1 the AMO spectral measures are singular continuous for every irrational α\alpha and every θ\theta. That conjecture is now a theorem (see reference 9).

Problem Statement

The surviving open core of the source's subcritical program, in its original scope, is question (A) below; question (B) is a further structural point that the source itself recorded as unresolved and that the subsequent literature has not addressed. The two questions are independent.

(A) Every-phase absolute continuity in the exponential regime for general potentials. Let v:R/Z→Rv:\mathbb{R}/\mathbb{Z}\to\mathbb{R} be real analytic and small in the sense of the source (the dual family {H^v,α,θ}θ∈R\{\hat H_{v,\alpha,\theta}\}_{\theta\in\mathbb{R}} is almost localized; in particular, by Theorem 5.1 of Avila–Jitomirskaya, λv\lambda v is small for ∣λ∣<λ0(v)|\lambda|<\lambda_0(v) at least when β(α)=0\beta(\alpha)=0). The source proved (Theorem 3.2) that if vv is small and β(α)=0\beta(\alpha)=0, then the spectral measures of Hv,α,θH_{v,\alpha,\theta} are absolutely continuous. Prove or disprove the extension to arbitrary irrational frequencies: if vv is small and α\alpha is exponentially Liouville, β(α)>0\beta(\alpha)>0, must the spectral measures of Hv,α,θH_{v,\alpha,\theta} be purely absolutely continuous for every phase θ∈R/Z\theta\in\mathbb{R}/\mathbb{Z}? A disproof consists of an explicit small analytic vv, an irrational α\alpha with β(α)>0\beta(\alpha)>0, and a phase θ\theta for which the spectral measure has a nonzero singular component (point or singular continuous). The target carries no arithmetic restriction on α\alpha, no restriction of vv to trigonometric polynomials, no loss of the every-phase quantifier, and no hypothesis tied to β\beta, such as an analyticity-strip width or a coupling threshold depending on β\beta (as in the unpublished Liu–Yuan claim).

(B) Structure of exceptional dual phases. In the quantitative Aubry duality used by the source, each energy typically corresponds to finitely many dual phases, but the source could not rule out uncountably many on an exceptional energy set (of Hausdorff dimension zero in its analysis), an issue the source links to the possible coexistence of point and singular continuous spectrum in the dual model. Settle this structural question for analytic almost-localized dual families: either rule out uncountably many dual phases on the exceptional energy set, or exhibit an energy at which uncountably many dual phases occur.

The common scientific objective is the source's subcritical program: a complete, arithmetic-free, every-phase description of absolutely continuous spectrum for one-frequency analytic quasiperiodic Schrödinger operators, of which the AMO Main Theorem is the model case. The source does not present (B) as an obstruction to (A): its footnote records the dual-phase multiplicity point as unresolved but harmless for the paper's own argument, since the exceptional set has Hausdorff dimension zero in its analysis, and its Section 1.3 identifies the obstacle to the exponential-regime generalization as the AMO-specific precise estimates ('the analysis of the exponential regime here is still heavily bound by precise estimates which are only available for the almost Mathieu operator, and a different analytic approach will need to be developed'). Relatedly, for the AMO the dual phase is explicit, which is what allowed the every-phase argument to be carried through all irrational frequencies; for general vv it is not.

References

  1. Artur Avila. The absolutely continuous spectrum of the almost Mathieu operator. Preprint, arXiv:0810.2965 (2008). DOI: 10.48550/arXiv.0810.2965. URL: https://arxiv.org/abs/0810.2965.
  2. Svetlana Ya. Jitomirskaya. Metal-insulator transition for the almost Mathieu operator. Annals of Mathematics (2) 150 (1999), no. 3, 1159–1175. DOI: 10.2307/121066. Preprint: arXiv:math/9911265. URL: https://doi.org/10.2307/121066
  3. Artur Avila, Svetlana Jitomirskaya. Almost localization and almost reducibility. Journal of the European Mathematical Society 12 (2010), no. 1, 93–131. DOI: 10.4171/jems/191. URL: https://doi.org/10.4171/jems/191
  4. Artur Avila. Almost reducibility and absolute continuity I. Preprint, arXiv:1006.0704 (2010). DOI: 10.48550/arXiv.1006.0704. URL: https://arxiv.org/abs/1006.0704
  5. Artur Avila. On point spectrum with critical coupling. Unpublished preprint (2008); the work has no DOI or arXiv identifier. Listed as 'Not intended for publication' on the author's publications page; cited as a preprint by Han–Schlag in their study of Avila's almost reducibility conjecture (see the Han–Schlag entry in this reference list). URL: https://w3.impa.br/~avila/papers.html
  6. Artur Avila. Global theory of one-frequency Schrödinger operators. Acta Mathematica 215 (2015), no. 1, 1–54. DOI: 10.1007/s11511-015-0128-7. URL: https://doi.org/10.1007/s11511-015-0128-7
  7. A. Avila, S. Jitomirskaya, C. A. Marx. Spectral theory of extended Harper's model and a question by Erdős and Szekeres. Inventiones mathematicae 210 (2017), no. 1, 283–339. DOI: 10.1007/s00222-017-0729-1. Preprint: arXiv:1602.05111. URL: https://doi.org/10.1007/s00222-017-0729-1
  8. Artur Avila, Raphaël Krikorian. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Annals of Mathematics (2) 164 (2006), no. 3, 911–940. DOI: 10.4007/annals.2006.164.911. Preprint: arXiv:math/0306382. URL: https://doi.org/10.4007/annals.2006.164.911
  9. Svetlana Jitomirskaya. On point spectrum of critical almost Mathieu operators. Advances in Mathematics 392 (2021) 107997. DOI: 10.1016/j.aim.2021.107997. URL: https://doi.org/10.1016/j.aim.2021.107997
  10. Svetlana Jitomirskaya, Igor Krasovsky. Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum. Preprint, arXiv:1909.04429 (2019). DOI: 10.48550/arXiv.1909.04429. URL: https://arxiv.org/abs/1909.04429
  11. Lingrui Ge, Svetlana Jitomirskaya, Xin Zhao. Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states. Communications in Mathematical Physics 402 (2023), no. 1, 213–229. DOI: 10.1007/s00220-023-04724-7. Preprint: arXiv:2204.11000. URL: https://doi.org/10.1007/s00220-023-04724-7
  12. Lingrui Ge. On the almost reducibility conjecture. Geometric and Functional Analysis 34 (2024), no. 1, 32–59. DOI: 10.1007/s00039-024-00671-0. Preprint: arXiv:2306.16376. URL: https://doi.org/10.1007/s00039-024-00671-0
  13. Rui Han, Wilhelm Schlag. Non-perturbative localization on the strip and Avila's almost reducibility conjecture. Preprint, arXiv:2306.15122 (2023). DOI: 10.48550/arXiv.2306.15122. URL: https://arxiv.org/abs/2306.15122
  14. Lingrui Ge, Svetlana Jitomirskaya, Jiangong You, Qi Zhou. Multiplicative Jensen's formula and quantitative global theory of one-frequency Schrödinger operators. Forum of Mathematics, Pi 14 (2026), e1. DOI: 10.1017/fmp.2025.10019. Preprint: arXiv:2306.16387. URL: https://doi.org/10.1017/fmp.2025.10019
  15. Wencai Liu. Small denominators and large numerators of quasiperiodic Schrödinger operators. Peking Mathematical Journal 8 (2025), no. 3, 503–532. DOI: 10.1007/s42543-023-00075-3. Preprint: arXiv:2205.04648. URL: https://doi.org/10.1007/s42543-023-00075-3
  16. Wencai Liu, Xiaoping Yuan. Absolutely continuous spectrum for the quasi-periodic Schrödinger operator in exponential regime. Preprint, arXiv:1311.0863 (2013). DOI: 10.48550/arXiv.1311.0863. URL: https://arxiv.org/abs/1311.0863
  17. Wencai Liu, Xiaoping Yuan. Hölder continuity of the spectral measures for one-dimensional Schrödinger operator in exponential regime. Journal of Mathematical Physics 56 (2015), no. 1, 012701. DOI: 10.1063/1.4904835. Preprint: arXiv:1311.0669. URL: https://doi.org/10.1063/1.4904835
  18. Wencai Liu, Xiaoping Yuan. Spectral gaps of almost Mathieu operators in the exponential regime. Journal of Fractal Geometry 2 (2015), no. 1, 1–51. DOI: 10.4171/jfg/15. URL: https://doi.org/10.4171/jfg/15
  19. David Damanik. The spectrum of the almost Mathieu operator. Survey notes, preprint arXiv:0908.1093 (2009). DOI: 10.48550/arXiv.0908.1093. URL: https://arxiv.org/abs/0908.1093
  20. Rui Han, Wilhelm Schlag. Hölder continuity of the integrated density of states for Liouville frequencies. Journal of Spectral Theory 16 (2026), no. 3, 1081–1108. DOI: 10.4171/jst/591. Preprint: arXiv:2408.15962. URL: https://doi.org/10.4171/jst/591