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Finite and full scale localization for the multi-frequency quasi-periodic CMV matrices

Bei Zhang, Daxiong Piao

TL;DR

This paper develops finite-scale and full-scale localization for multi-frequency quasi-periodic CMV matrices, extending the localization framework known for MF-QP Schrödinger operators to the CMV setting. The authors establish a robust Lyapunov-Exponential structure via large deviation estimates for the monodromy and its entries, derive a CMV-specific Poisson formula, and prove finite-scale localization by combining LDT with the Avalanche Principle and Poisson analysis. They then introduce a semialgebraic-sets approach and a no-double-resonance (NDR) condition to eliminate double resonances, culminating in full-scale localization through scale-bridging arguments, Cartan-type control, and Weierstrass preparation. The results significantly advance spectral localization theory for MF-QP CMV matrices, connecting CMV analysis with the established MF-QP Schrödinger theory and impacting quantum-walk/CMV-related spectral problems.

Abstract

This paper formulates the finite and full-scale localization for multi-frequency quasi-periodic CMV matrices. This can be viewed as the CMV counterpart to the localization results by Goldstein, Schlag, and Voda [arXiv:1610.00380 (math.SP], Invent. Math. 217 (2019)) on multi-frequency quasi-periodic Schrödinger operators.

Finite and full scale localization for the multi-frequency quasi-periodic CMV matrices

TL;DR

This paper develops finite-scale and full-scale localization for multi-frequency quasi-periodic CMV matrices, extending the localization framework known for MF-QP Schrödinger operators to the CMV setting. The authors establish a robust Lyapunov-Exponential structure via large deviation estimates for the monodromy and its entries, derive a CMV-specific Poisson formula, and prove finite-scale localization by combining LDT with the Avalanche Principle and Poisson analysis. They then introduce a semialgebraic-sets approach and a no-double-resonance (NDR) condition to eliminate double resonances, culminating in full-scale localization through scale-bridging arguments, Cartan-type control, and Weierstrass preparation. The results significantly advance spectral localization theory for MF-QP CMV matrices, connecting CMV analysis with the established MF-QP Schrödinger theory and impacting quantum-walk/CMV-related spectral problems.

Abstract

This paper formulates the finite and full-scale localization for multi-frequency quasi-periodic CMV matrices. This can be viewed as the CMV counterpart to the localization results by Goldstein, Schlag, and Voda [arXiv:1610.00380 (math.SP], Invent. Math. 217 (2019)) on multi-frequency quasi-periodic Schrödinger operators.

Paper Structure

This paper contains 16 sections, 56 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Useful Lemmas
  4. LDT Estimate for the Monodromy
  5. Estimation of the Lyapunov Exponent
  6. LDT Estimate for the Entries of the Monodromy
  7. Poisson Formula for CMV Matrix
  8. Finite Scale Localization
  9. Removing Double Resonances by Using Semialgebraic Sets
  10. Removing Double Resonances under NDR Condition
  11. Full Scale Localization
  12. Appendix
  13. Some Useful Notations
  14. Avalanche Principle
  15. Subharmonic Functions
  16. ...and 1 more sections

Key Result

Lemma 3.1

GS01-Annals Let $d$ be a positive integer. Suppose $u: D(0,2)^{d}\rightarrow [-1,1]$ is subharmonic in each variable; i.e., $z\rightarrow u(z_{1},u_{2},\ldots,z_{d})$ is subharmonic for any choice of $(z_{2},\ldots,z_{d})\in D(0,2)^{d-1}$ and similarly for each of the other variables. Assume further Then there exist $\sigma>0$, $\tau>0$, and $c_{0}$ only depending on $d$ and $\varepsilon_{1}$ such

Theorems & Definitions (100)

  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 90 more