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Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys

Lingrui Ge, Jiangong You, Qi Zhou

TL;DR

The paper develops a comprehensive, arithmetic-sensitive framework—structured quantitative almost reducibility—to resolve long-standing questions about sharp spectral gaps, localization, and reducibility for analytic quasiperiodic Schrödinger operators. By handling infinitely many rotation-number resonances and refining Aubry duality, it achieves optimal exponential gap decay for the AMO, sharp arithmetic Anderson localization in multiple dimensions, and stratified growth patterns of extended eigenfunctions. The approach unifies local and global reducibility analyses, proves sharp phase-transition phenomena, and extends these insights to a broad class of type-I operators via a refined duality theory. Overall, the work advances both the dynamical and spectral understanding of quasiperiodic systems, with implications for Cantor-spectrum structure and quantum transport in quasi-crystals.

Abstract

This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators. The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics.

Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys

TL;DR

The paper develops a comprehensive, arithmetic-sensitive framework—structured quantitative almost reducibility—to resolve long-standing questions about sharp spectral gaps, localization, and reducibility for analytic quasiperiodic Schrödinger operators. By handling infinitely many rotation-number resonances and refining Aubry duality, it achieves optimal exponential gap decay for the AMO, sharp arithmetic Anderson localization in multiple dimensions, and stratified growth patterns of extended eigenfunctions. The approach unifies local and global reducibility analyses, proves sharp phase-transition phenomena, and extends these insights to a broad class of type-I operators via a refined duality theory. Overall, the work advances both the dynamical and spectral understanding of quasiperiodic systems, with implications for Cantor-spectrum structure and quantum transport in quasi-crystals.

Abstract

This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators. The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics.
Paper Structure (31 sections, 38 theorems, 380 equations)

This paper contains 31 sections, 38 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Exact Exponential Decay Rate of the Spectral Gaps
  3. Optimal arithmetic Anderson localization and optimal reducibility
  4. Optimal arithmetic Anderson localization
  5. Sharp phase trasition
  6. Optimal arithmetic reducibility
  7. Stratified, optimal asymptotic growth of extended eigenfunction
  8. Difficulty, Novelty, ideas of the proof
  9. Infinitely many rotation number resonances
  10. Structured quantitative almost reducibility
  11. Sharp version of quantitative Aubry duality.
  12. Preliminaries
  13. Quasiperiodic cocycles
  14. Almost Mathieu cocycle
  15. Aubry duality
  16. ...and 16 more sections

Key Result

Theorem 1.1

Assume $\alpha \in \text{DC}$ and $|\lambda| \neq 1$, then we have

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.1
  • Theorem 1.6
  • ...and 38 more