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

TL;DR

This paper investigates the regularity of the Lyapunov exponent for $C^2$ cos-type quasiperiodic Schr"odinger cocycles with large couplings, establishing absolute continuity and precise H"older continuity properties.

Contribution

It proves the Lyapunov exponent is $rac{1}{2}$-H"older continuous and characterizes local regularity variations within the spectrum for large coupling quasiperiodic Schr"odinger cocycles.

Findings

Lyapunov exponent is $rac{1}{2}$-H"older continuous.

Absolute continuity of the Lyapunov exponent.

Existence of energies with variable local regularity in the spectrum.

Abstract

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schr\"odinger 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 $\frac{1}{2}$-H\"older 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 $(r-\epsilon)$-H\"older continuity and $(r+\epsilon)$-H\"older continuity.