[Paper Review] Almost localization and almost reducibility

Artur Avila, Svetlana Jitomirskaya|ArXiv.org|May 12, 2008

Spectral Theory in Mathematical Physics16 references3 citations

TL;DR

This paper develops a quantitative version of Aubry duality to establish almost reducibility and full localization for quasiperiodic Schrödinger operators with small analytic potentials and Diophantine frequencies, proving 1/2-Hölder continuity of the integrated density of states and resolving the dry version of the Ten Martini Problem in the non-perturbative regime.

ABSTRACT

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson's reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, ``almost reducibility'' in some band of analyticity). We also prove 1/2-Hölder continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of sub-critical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters.

Motivation & Objective

• To overcome the limitations of prior duality methods that lacked quantitative estimates, particularly for energies where localization or reducibility fails.
• To extend Eliasson's perturbative reducibility theory to the non-perturbative regime, valid for all Diophantine frequencies and all energies in the spectrum.
• To establish 1/2-Hölder continuity of the integrated density of states, a key spectral regularity property.
• To resolve the dry version of the Ten Martini Problem for the almost Mathieu operator in the entire sub-critical coupling regime.
• To provide a framework for analyzing the full spectrum, including zero-measure sets of 'bad' energies previously inaccessible to algebraic duality.

Proposed method

• Develop a quantitative version of Aubry duality that links spectral properties of the dual model (localization) to dynamical properties of the original model (reducibility).
• Use Rouche's Theorem and degree bounds on trigonometric polynomials to control the growth of transfer matrices and derive uniform estimates.
• Apply a KAM-type scheme with improved estimates to achieve almost reducibility for all energies, even when full reducibility fails.
• Establish bounds on the norm of transfer matrices via control of the degree of associated trigonometric polynomials under perturbation.
• Use cohomological equation solving and conjugation techniques to reduce the cocycle to a near-constant form with exponentially small error.
• Leverage the Diophantine condition on the frequency to control small divisors and ensure convergence of iterative schemes.

Experimental results

Research questions

• RQ1 Can a quantitative duality theory be developed to analyze the full spectrum of quasiperiodic Schrödinger operators, including energies where localization or reducibility fails?
• RQ2 Can Eliasson's perturbative reducibility results be extended to the non-perturbative regime, independent of the frequency's Diophantine properties?
• RQ3 What is the optimal modulus of continuity of the integrated density of states for small analytic potentials?
• RQ4 Does the dry version of the Ten Martini Problem hold for the almost Mathieu operator in the entire sub-critical coupling regime?
• RQ5 Can almost reducibility be established uniformly across all energies, even in the absence of full reducibility?

Key findings

• The authors establish 1/2-Hölder continuity of the integrated density of states for quasiperiodic Schrödinger operators with small analytic potentials and Diophantine frequencies.
• They prove almost reducibility for all energies in the spectrum, even when full reducibility fails, by developing a quantitative duality framework.
• The full version of Eliasson's reducibility theory is extended to the non-perturbative regime, valid for all Diophantine frequencies.
• For the almost Mathieu operator, the results hold throughout the entire sub-critical coupling regime, resolving the dry version of the Ten Martini Problem.
• The method yields improved estimates in the perturbative regime, showing almost reducibility holds in a band of analyticity for all energies.
• The degree of transfer matrices is bounded by a linear function of the inverse of the analyticity radius, enabling uniform control via Rouche's Theorem.