  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.

Dynamical Systems