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

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