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Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: II. Large Deviations and Anderson Localization

Artur Avila, David Damanik, Zhenghe Zhang

Abstract

We consider discrete one-dimensional Schrödinger operators whose potentials are generated by Hölder continuous sampling along the orbits of a uniformly hyperbolic transformation. For any ergodic measure satisfying a suitable bounded distortion property, we establish a uniform large deviation estimate in a large energy region provided that the sampling function is locally constant or has small supremum norm. We also prove full spectral Anderson localization for the operators in question.

Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: II. Large Deviations and Anderson Localization

Abstract

We consider discrete one-dimensional Schrödinger operators whose potentials are generated by Hölder continuous sampling along the orbits of a uniformly hyperbolic transformation. For any ergodic measure satisfying a suitable bounded distortion property, we establish a uniform large deviation estimate in a large energy region provided that the sampling function is locally constant or has small supremum norm. We also prove full spectral Anderson localization for the operators in question.
Paper Structure (14 sections, 19 theorems, 208 equations)

This paper contains 14 sections, 19 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. The Setting and the Main Results
  3. The Base Dynamical System
  4. Measures With the Bounded Distortion Property
  5. The Associated Operator Family
  6. The Main Results
  7. Large Deviation Estimates -- Proof of Theorem \ref{['t:uld']}
  8. Stable and Unstable Holonomies
  9. Reduction of the Uniform LDT
  10. Proof of Theorem \ref{['t:ldt_additive']}
  11. Establishing Localization -- Proof of Theorem \ref{['t:main']}
  12. Bounding the Green Function
  13. Elimination of Double Resonances
  14. Applications -- Proof of Corollaries \ref{['t:locally_constant']} and \ref{['t:fiber_bunched']}

Key Result

Theorem 2.10

Let $\Omega$ be a subshift of finite type and $\mu$ be a $T$-ergodic measure that has the bounded distortion property. Assume that $T$ has a fixed point. Let $f\in {\mathrm {LC}} \cup {\mathrm {SH}}$ be nonconstant. Then there is a connected compact interval $J\supset\Sigma_f$ such that $A^{E}$ sati

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 39 more