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Anderson localization and Hölder regularity of IDS for analytic quasi-periodic Schrödinger operators

Hongyi Cao, Yunfeng Shi, Zhifei Zhang

Abstract

We establish in the perturbative regime both Anderson localization and Hölder continuity of IDS for quasi-periodic Schrödinger operators on $\mathbb{Z}^d$ with any non-constant analytic potentials and fixed Diophantine frequencies. Our approach provides a new way to control Green's functions in the spirit of multi-scale analysis.

Anderson localization and Hölder regularity of IDS for analytic quasi-periodic Schrödinger operators

Abstract

We establish in the perturbative regime both Anderson localization and Hölder continuity of IDS for quasi-periodic Schrödinger operators on with any non-constant analytic potentials and fixed Diophantine frequencies. Our approach provides a new way to control Green's functions in the spirit of multi-scale analysis.
Paper Structure (29 sections, 43 theorems, 242 equations)

This paper contains 29 sections, 43 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Main ideas and new ingredients of the proof
  4. Extension of Bou00's argument beyond cosine potential
  5. Weierstrass preparation argument for the $E$-variable
  6. Elimination of double resonances via transversality of the resultant
  7. Organization of the paper
  8. Some notation in the paper
  9. Multi-scale analysis
  10. The initial step
  11. Green's function estimates for $0$-nonresonant sets
  12. The first inductive step
  13. Choice of scale
  14. Resonance analysis via Schur complement
  15. Green's function estimates for $1$-nonresonant sets
  16. ...and 14 more sections

Key Result

Theorem 1.3

Assume that $\omega\in {\rm DC}_{\tau, \gamma}$ and $v(\theta)$ satisfies tiao1, Conditions con1 and con2. Then there exists some $\varepsilon_0=\varepsilon_0(\eta,C_v,M, \widetilde{C}_v, c, s ,d,\tau,\gamma)>0$ such that for all $0\leq \varepsilon\leq \varepsilon_0$, $H(\theta)$ satisfies the And

Theorems & Definitions (82)

  • Theorem 1.3
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 72 more