Title: Non-perturbative positive Lyapunov exponent of Schrödinger equations and its applications to skew-shift,
Abstract: We first study the discrete Schrödinger equations with analytic potentials
given by a class of transformations. It is shown that if the coupling number is
large, then its logarithm equals approximately to the Lyapunov exponents. When
the transformation becomes the skew-shift, we prove that the Lyapunov exponent
is week Hölder continuous, and the spectrum satisfies Anderson Localization
and contains large intervals. Moreover, all of these conclusions are
non-perturbative.