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Explicit construction of gevrey quasi-periodic discrete schrödinger operators with cantor spectrum

Xuanji Hou, Li Zhang

TL;DR

The paper provides an explicit construction of a Gevrey quasi-periodic potential for a one-dimensional discrete Schrödinger operator that yields Cantor spectrum with open spectral gaps. The approach combines a carefully designed KAM scheme for Schrödinger cocycles with a Moser–Pöschel argument to guarantee gap openings at a dense set of labels $\mathcal{K}$ and to obtain quantitative gap bounds. A key contribution is the explicit construction of the resonance set $\mathcal{K}$ and the control of reducibility toward hyperbolic or parabolic constant cocycles, which in turn ensures dense spectral gaps and a Cantor spectrum for all $0<|\lambda|\le 1$. The work advances explicit, nonperturbative spectral results for quasi-periodic operators with Gevrey regularity, providing concrete gap-size estimates and bridging dynamical KAM methods with spectral theory.

Abstract

We construct 1-dim difference Schrödinger operators with a class of Gevrey potentials such that Cantor spectrum occurs together with the estimations of open spectral gaps . The proof is based on KAM and Moser-Pöschel argument .

Explicit construction of gevrey quasi-periodic discrete schrödinger operators with cantor spectrum

TL;DR

The paper provides an explicit construction of a Gevrey quasi-periodic potential for a one-dimensional discrete Schrödinger operator that yields Cantor spectrum with open spectral gaps. The approach combines a carefully designed KAM scheme for Schrödinger cocycles with a Moser–Pöschel argument to guarantee gap openings at a dense set of labels and to obtain quantitative gap bounds. A key contribution is the explicit construction of the resonance set and the control of reducibility toward hyperbolic or parabolic constant cocycles, which in turn ensures dense spectral gaps and a Cantor spectrum for all . The work advances explicit, nonperturbative spectral results for quasi-periodic operators with Gevrey regularity, providing concrete gap-size estimates and bridging dynamical KAM methods with spectral theory.

Abstract

We construct 1-dim difference Schrödinger operators with a class of Gevrey potentials such that Cantor spectrum occurs together with the estimations of open spectral gaps . The proof is based on KAM and Moser-Pöschel argument .
Paper Structure (20 sections, 24 theorems, 295 equations)

This paper contains 20 sections, 24 theorems, 295 equations.

Table of Contents

  1. INTRODUCTION AND MAIN RESULTS
  2. Cantor Spectrum Of Quasi-Periodic Schrödinger Operators
  3. Estimates On Spectral Gap
  4. Structure of the paper
  5. PRELIMINARIES
  6. Quasi-periodic Cocycles And Fibered Rotation Number
  7. On Functions
  8. On Algebraic Conjugations
  9. Continued Fraction Expansion
  10. Construction Of $\mathcal{K}$
  11. KAM PROPOSITION
  12. One Step Of KAM
  13. KAM Iteration Lemma
  14. PROOF OF THEOREM \ref{['th-1.2']}
  15. More Estimates
  16. ...and 5 more sections

Key Result

Theorem 1.1

Given $s\in(0,\frac{1}{2})$ and $\alpha\in {\rm DC}_d(\gamma,\tau)$. One can construct explicitly a set $\mathcal{K}\subseteq\mathbb{Z}^{d}$ depending on $\alpha$ and $s$, such that for the Schrödinger operator $H_{\lambda v,\alpha,\theta}$1.1 with is a Gevrey real function, $\Sigma_{\lambda v,\alpha}$ is Cantor for all $0<|\lambda|\leq1$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 15 more