H\"older continuity of Lyapunov exponent for a family of smooth Schr\"odinger cocycles

TL;DR

This paper proves that the Lyapunov exponent for certain smooth quasi-periodic Schr"odinger cocycles is H"older continuous, with a universal exponent independent of frequency and coupling strength, under large coupling conditions.

Contribution

It establishes the H"older continuity of the Lyapunov exponent for smooth Schr"odinger cocycles with Liouvillean frequencies, extending previous results to a broader class of potentials.

Findings

Lyapunov exponent is H"older continuous for large coupling

H"older exponent is independent of frequency and coupling

Results apply to $C^2$ cos-type potentials with Liouvillean frequencies

Abstract

We prove the H\"older continuity of the Lyapunov exponent for quasi-periodic Schr\"odinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the H\"older exponent is independent of the frequency and the coupling constant.