Exact local distribution and sharp arithmetic thresholds for absolutely continuous spectral measures of one-frequency quasiperiodic Schrödinger operators at subcritical energies

Avila and Jitomirskaya (Commun. Math. Phys. 301 (2011) 563–581) proved that for one-frequency analytic quasiperiodic Schrödinger operators with Diophantine frequency, the individual absolutely continuous spectral measures are 1/2-Hölder continuous at almost reducible energies, uniformly in the phase, and left open several questions: the Almost Reducibility Conjecture and the description of the almost-reducible summand, extensions of the estimates beyond Diophantine frequencies, and quantitative control of the density (modulus of absolute continuity) of the spectral measures. The structural questions have been settled by Avila's global theory and 2023 spectral-dichotomy work (subcriticality implies almost reducibility; the typical operator has no singular continuous spectrum), and the exact local distribution of the absolutely continuous spectral measure is now known for every subcritical almost Mathieu operator with Diophantine frequency (Li–You–Zhou, Proc. Lond. Math. Soc. 132 (2026)): local dimension 1 except at countably many gap-edge energies with dimension 1/2, with two-sided fractal asymptotics, its multifractal spectrum completed in 2025. The surviving open core has a common objective — completing the fine quantitative description of the individual absolutely continuous spectral measures at subcritical energies beyond the almost-Mathieu/Diophantine case: (i) the exact local distribution (local dimensions with matching lower bounds) for general analytic potentials, where only stratified one-sided upper bounds are known and the almost-Mathieu picture is known to fail in general, together with the quantitative density bounds (whether the measure of every Borel set is at most a constant times the square root of its Lebesgue measure); and (ii) the sharp arithmetic condition on the frequency separating the regime where the fine estimates hold from the extremely well-approximated frequencies where Hölder regularity fails.

Background

A discrete Schrödinger operator with a quasiperiodic potential acts on $\ell^2(\mathbb{Z})$ by $(Hu)_n = u_{n+1}+u_{n-1}+v(\theta+n\alpha)u_n,$ where $v$ is a fixed potential on the circle $\mathbb{R}/\mathbb{Z}$, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ is the frequency, and $\theta$ is the phase; the central example, the almost Mathieu operator (AMO), has $v(x)=2\lambda\cos(2\pi x)$. For a vector $f\in\ell^2(\mathbb{Z})$, the spectral measure $\mu^f$ is defined by $\int\varphi\,d\mu^f=\langle\varphi(H)f,f\rangle$ for bounded Borel functions $\varphi$; its fine regularity, such as Hölder continuity of its distribution function, encodes how spectral weight is distributed and constrains quantum transport (spreading of wavepackets, return probabilities). Averaging the measures $\mu^f$ over the phase yields the integrated density of states (IDS), the distribution function $N(E)$ of a phase-averaged measure, whose regularity is much easier to analyze. Absolutely continuous (ac) spectrum — the regime where the spectral measure has an $L^1$ density with respect to Lebesgue measure — is exceptional among one-dimensional ergodic operators, but for analytic $v$ it occurs throughout the small-coupling regime (potentials that are small perturbations of zero, the coupling being the size of $v$) and for the AMO throughout the subcritical regime $|\lambda|<1$.

The dynamical reformulation associates to each energy $E$ the Schrödinger cocycle $(\alpha,A^{(E-v)})$ with $A^{(E-v)}(x)=\begin{pmatrix}E-v(x)&-1\\1&0\end{pmatrix}$, whose iterates $A_n$ govern solutions of the eigenvalue equation. Its Lyapunov exponent is $L(E)=\lim_{n\to\infty}\frac{1}{n}\int\ln\|A_n(x)\|dx$. An energy $E$ is called subcritical if $\ln\|A_n(x)\|=o(n)$ uniformly in some complex band $|\Im x|<h$; the cocycle is almost reducible if it is analytically conjugate to cocycles arbitrarily close to constants. By the renormalization theory of Avila and Krikorian (Ann. of Math. 164 (2006) 911–940), almost reducible energies form an essential support of the ac spectrum. A frequency is Diophantine if its continued-fraction denominators $q_n$ grow at most geometrically ($\ln q_{n+1}/\ln q_n=O(1)$); Liouville (well-approximated) frequencies fail this.

The source paper (Avila and Jitomirskaya, Commun. Math. Phys. 301 (2011) 563–581) proved that for Diophantine $\alpha$ and analytic $v$, at almost reducible energies, each individual ac spectral measure is 1/2-Hölder on intervals, $\mu^f_{v,\alpha,\theta}(J)\le C|J|^{1/2}\|f\|^2_{\ell^1}$, uniformly in the phase — the first fine bounds for individual ac measures of ergodic operators. The exponent is sharp because of square-root (van Hove-type) singularities of the density at gap edges of the spectrum. The paper posed the Almost Reducibility Conjecture (ARC), that subcriticality implies almost reducibility, as the key to its Spectral Dichotomy Conjecture: for typical (in the measure-theoretic sense of prevalence) $v,\alpha,\theta$, the operator splits as a direct sum $H_+\oplus H_-$ of a localized part and an almost reducible part with disjoint spectra. It also asked (Remark 1.3) to extend the phase-uniform bounds beyond Diophantine frequencies, to Brjuno or controlled exponential approximation, expecting the modulus to depend on the Diophantine properties of $\alpha$; and (Remark 4.4) to estimate the modulus of absolute continuity of the measures, suggesting that $\mu(X)\le C|X|^{1/2}$ may hold for all Borel sets $X$ (where $|X|$ is Lebesgue measure), a statement equivalent to weak-$L^2$ control of the ac density, with the unboundedness of the density caused exactly by countably many decaying square-root singularities at gap boundaries.

Current status (per the cited literature): the ARC is a theorem of Avila (preprints arXiv:1006.0704 (2010) and arXiv:2307.11071 (2023), together with the global theory in Acta Math. 215 (2015) 1–54), and it yields the Spectral Dichotomy for typical one-frequency operators (the typical operator has no singular continuous spectrum; $H_-$ is the restriction to the subcritical energies). Liu and Yuan (J. Math. Phys. 56 (2015) 012701) extended the 1/2-Hölder continuity to all frequencies of finite exponential type $\beta(\alpha)<\infty$, at the cost of small coupling, while Avila, Last, Shamis and Zhou (Duke Math. J. 173 (2024) 603–672) showed that for extremely well-approximated frequencies even the IDS is not Hölder. Li, You and Zhou (Proc. Lond. Math. Soc. 132 (2026)) determined the exact local distribution of the ac spectral measure for every subcritical AMO with Diophantine frequency: the local dimension $D_\mu(E)=\lim_{\epsilon\to0}\frac{\ln\mu(E-\epsilon,E+\epsilon)}{\ln\epsilon}$ equals 1 except at the countably many energies whose IDS value is resonant ($N(E)=k\alpha\bmod\mathbb{Z}$, the gap labels), where it equals 1/2, with two-sided asymptotics governed by resonances of the IDS; this answers the 1986 physics conjecture of Tang and Kohmoto, and the multifractal spectrum was completed by Cao, Li, Wang and Zhou (arXiv:2509.09945 (2025)). For general analytic potentials, only stratified one-sided Hölder upper bounds are known, and the AMO dimension picture is known to fail in general (collapsed gaps; reducible cocycles give Lipschitz measures). What remains open is stated in the problem statement.

Problem Statement

Let $H_{v,\alpha,\theta}$ be the one-frequency quasiperiodic Schrödinger operator on $\ell^2(\mathbb{Z})$, $(Hu)_n=u_{n+1}+u_{n-1}+v(\theta+n\alpha)u_n,$ with $v\in C^\omega(\mathbb{R}/\mathbb{Z},\mathbb{R})$ real-analytic, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$, $\theta\in\mathbb{R}$, and let $\mu^f_{v,\alpha,\theta}$ be the spectral measure of $f\in\ell^1(\mathbb{Z})$. Call an energy $E$ in the spectrum subcritical if the Schrödinger cocycle $(\alpha,A^{(E-v)})$ satisfies $\ln\|A_n(x)\|=o(n)$ uniformly in some band $|\Im x|<h$; subcritical energies support the absolutely continuous spectrum and are now known to be almost reducible. The single scientific objective of this problem is to complete the fine quantitative description of the individual absolutely continuous spectral measures at subcritical energies — the description initiated by Avila–Jitomirskaya's 1/2-Hölder theorem and completed for the almost Mathieu operator — in the two directions that remain open:

(1) General analytic potentials (Diophantine $\alpha$). Determine the exact local distribution of $\mu^f_{v,\alpha,\theta}$ at subcritical energies for a general analytic potential $v$: compute the local dimension $D_{\mu^f}(E)=\lim_{\epsilon\to0}\frac{\ln\mu^f_{v,\alpha,\theta}(E-\epsilon,E+\epsilon)}{\ln\epsilon}$ (or the upper and lower local dimensions, or two-sided asymptotics of $\mu^f(E-\epsilon,E+\epsilon)$ as $\epsilon\to0$) at each energy (the dependence on the vector $f$, if any, should be addressed), with matching lower bounds — currently only stratified one-sided Hölder upper bounds are known (upper bounds whose exponent depends on resonance parameters of the energy relative to the frequency), and the almost-Mathieu picture (dimension 1 except countably many gap-edge square-root singularities of dimension 1/2) is known to fail in general (collapsed gaps; reducible cocycles give Lipschitz behavior). As part of this, settle the quantitative density control asked in Remark 4.4 of the source paper: does the modulus of absolute continuity bound $\mu^{\mathrm{ac}}(X)\le C|X|^{1/2}$ hold for all Borel sets $X$ (where $|X|$ is Lebesgue measure, $\mu^{\mathrm{ac}}$ is the absolutely continuous part of the spectral measure, and the source's Remark 4.4 states this for the full spectral measure), with $C$ uniform in the phase $\theta$ — equivalently, weak-$L^2$ control of the ac density — and more generally identify which density bounds hold at subcritical energies for general $v$?

(2) Sharp arithmetic condition on the frequency. Determine the exact class of irrational frequencies $\alpha$ for which these fine estimates hold at subcritical energies in the natural nonperturbative regime (e.g. for the AMO, all $|\lambda|<1$): characterize the boundary between frequencies of small exponential approximation rate $\beta(\alpha)=\limsup_{n\to\infty}\frac{\ln q_{n+1}}{q_n}<\infty$ (where 1/2-Hölder continuity is known, at least for small coupling) and extremely well-approximated frequencies (where Hölder regularity is known to fail), and describe how the optimal modulus of continuity depends on the Diophantine properties of $\alpha$.

The two parts are linked components of one objective: rigorous theorems identifying the local dimension or two-sided asymptotics (with lower bounds) for general potentials, and the sharp arithmetic class with positive results and counterexamples delimiting it.

References

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2. Artur Avila and Raphaël Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Annals of Mathematics (2) 164 (2006), 911–940. DOI: 10.4007/annals.2006.164.911, https://doi.org/10.4007/annals.2006.164.911
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