The Precise Regularity of the Lyapunov Exponent for \(C^2\) Cos-Type Quasiperiodic Schrödinger Cocycles with Large Couplings

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 \(r\in (\frac{1}{2}, 1)\), we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between \((r-\epsilon )\)-Hölder continuity and \((r+\epsilon )\)-Hölder continuity.