Sharp $\frac12$-H\"older continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schr\"odinger cocycles

TL;DR

This paper proves that the Lyapunov exponent for certain Schr"odinger cocycles exhibits sharp 2-H40lder continuity at the spectrum's bottom, especially in large coupling regimes with specific potential properties.

Contribution

It establishes the sharp 2-H40lder continuity of the Lyapunov exponent at the spectrum's bottom for Schr40dinger cocycles with a potential having a unique non-degenerate minimum.

Findings

Lyapunov exponent is 2-H40lder continuous at the spectrum's bottom.

Continuity is sharp, matching known upper bounds.

Results apply in the large coupling regime for potentials with specific minima.

Abstract

We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerkl\"ov and Saprykina (2008, Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $\frac12$-H\"older continuous. In particular, we show that the Lyapunov exponent of Schr\"odinger cocycles with a potential having a unique non-degenerate minimum, is sharp $\frac12$-H\"older continuous below the lowest energy of the spectrum, in the large coupling regime.