Acta Mathematica

Global theory of one-frequency Schrödinger operators

Artur Avila

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Abstract

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.

Key to our approach are two results about the dependence of the Lyapunov exponent of one-frequency SL(2,C) cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe “quantization” restriction, and on the other hand “regularity” of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency.

Note

I am grateful to Svetlana Jitomirskaya and David Damanik for several detailed comments which greatly improved the exposition. This work was partially conducted during the period the author served as a Clay Research Fellow. This work has been supported by the ERC Starting Grant “Quasiperiodic” and by the Balzan project of Jacob Palis.

Article information

Source
Acta Math., Volume 215, Number 1 (2015), 1-54.

Dates
Received: 8 April 2013
Revised: 15 June 2015
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802442

Digital Object Identifier
doi:10.1007/s11511-015-0128-7

Mathematical Reviews number (MathSciNet)
MR3413976

Zentralblatt MATH identifier
1360.37072

Subjects
Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D30: Partially hyperbolic systems and dominated splittings 37J10: Symplectic mappings, fixed points 37C20: Generic properties, structural stability

Keywords
symplectic diffeomorphisms partial hyperbolicity Lyapunov exponents generic properties

Rights
2015 © Institut Mittag-Leffler

Citation

Avila, Artur. Global theory of one-frequency Schrödinger operators. Acta Math. 215 (2015), no. 1, 1--54. doi:10.1007/s11511-015-0128-7. https://projecteuclid.org/euclid.acta/1485802442


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