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The precise regularity of the Lyapunov exponent for $C^2$ Cos-type quasiperiodic Schrödinger cocycles with large couplings

Jiahao Xu, Lingrui Ge, Yiqian Wang

TL;DR

The paper proves that for a class of quasi-periodic Schrödinger cocycles with $C^2$ cos-type potentials at large coupling and Diophantine frequency, the Lyapunov exponent $L(E)$ is absolutely continuous and $1/2$-Hölder on any compact energy interval. The authors introduce a resonance-aware inductive scheme and derive sharp bounds on the derivative of the finite Lyapunov exponent, leveraging a weak Large Deviation Theorem and the Avalanche Principle. They establish a detailed connection between resonance structure and spectral gaps, label gaps by a new index $k$ and show endpoints of gaps control endpoint regularity, with $L$ differentiable almost everywhere and a quantified tail bound on $|L'(E)|$. The core technique blends resonance geometry, controlled derivative estimates of FLE, and iterative scale analysis to obtain optimal regularity, including exact local $1/2$-Hölder regularity at spectral gap endpoints and absolute continuity across spectra. These results advance understanding of LE regularity in non-analytic, large-coupling regimes and have implications for spectral Cantor structure and localization phenomena in quasiperiodic operators.

Abstract

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with $C^2$ cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is $\frac{1}{2}$-Hölder continuous. Furthermore, for any given $r\in (\frac12, 1)$, we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between $(r-ε)$-Hölder continuity and $(r+ε)$-Hölder continuity.

The precise regularity of the Lyapunov exponent for $C^2$ Cos-type quasiperiodic Schrödinger cocycles with large couplings

TL;DR

The paper proves that for a class of quasi-periodic Schrödinger cocycles with cos-type potentials at large coupling and Diophantine frequency, the Lyapunov exponent is absolutely continuous and -Hölder on any compact energy interval. The authors introduce a resonance-aware inductive scheme and derive sharp bounds on the derivative of the finite Lyapunov exponent, leveraging a weak Large Deviation Theorem and the Avalanche Principle. They establish a detailed connection between resonance structure and spectral gaps, label gaps by a new index and show endpoints of gaps control endpoint regularity, with differentiable almost everywhere and a quantified tail bound on . The core technique blends resonance geometry, controlled derivative estimates of FLE, and iterative scale analysis to obtain optimal regularity, including exact local -Hölder regularity at spectral gap endpoints and absolute continuity across spectra. These results advance understanding of LE regularity in non-analytic, large-coupling regimes and have implications for spectral Cantor structure and localization phenomena in quasiperiodic operators.

Abstract

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is -Hölder continuous. Furthermore, for any given , we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between -Hölder continuity and -Hölder continuity.

Paper Structure

This paper contains 44 sections, 79 theorems, 1105 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Remarks on the regularity of the Lyapunov exponents
  4. The idea for the proof
  5. Definitions and Notations
  6. Preliminaries
  7. Some basic lemmas for the sharp estimate on the derivative of finite LE
  8. The induction Theorem for $C^2$ Cosine Type
  9. The Large Deviation Theorem and Avalanche Principle
  10. The resonance and the spectral gaps
  11. The proof of the main theorem based on a sharp estimate of the derivative on FLE (Lemma \ref{['lemma18']})
  12. A sharp estimate on the derivative of FLE
  13. Several useful inequalities
  14. The proof of Theorem \ref{['Th1']}
  15. Exactly local $\frac{1}{2}$-Hölder continuity for $t\in EP$
  16. ...and 29 more sections

Key Result

Theorem 2

Let $\alpha$ be Diophantine and $v$ be of $C^{2}$ cos-type. Consider the quasiperiodic Schrödinger cocycle $(\alpha, A^{(E-\lambda v)})$ and let $L(E)=L(E,\lambda)$ be its LE. Then there exists some $\lambda_1=\lambda_1(\alpha,v)>0$ such that for any fixed $\lambda>\lambda_1$, the followings hold tr

Theorems & Definitions (180)

  • Definition 1.1
  • Definition 1.2
  • Remark 1
  • Theorem 2
  • Corollary 3
  • Remark 4
  • Definition 1.3: Equivalent relation
  • Definition 1.4: Projection operator
  • Lemma 5: z1
  • Lemma 6
  • ...and 170 more