Sharp $\frac{1}{2}$-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Abstract

We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv 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 $\frac{1}{2}$-Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp $\frac{1}{2}$-Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.