Non-perturbative positive Lyapunov exponent of Schrodinger equations and its applications to skew-shift mapping

摘要

We first study the discrete Schrodinger equations with analytic potentials given by a class of transformations. It is shown that if the coupling number is large, then the Lyapunov exponent equals approximately to the logarithm of this coupling number. When the transformation becomes the skew-shift mapping, we prove that the Lyapunov exponent is weak Holder continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative. (C) 2018 Elsevier Inc. All rights reserved.