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arXiv · 1712.07900

Non-perturbative positive Lyapunov exponent of Schr\"odinger equations and its applications to skew-shift

Kai Tao
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Abstract

We first study the discrete Schr\"odinger 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 H\"older continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative.

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Kai Tao. 2017-12-21. Non-perturbative positive Lyapunov exponent of Schr\"odinger equations and its applications to skew-shift. https://arxiv.org/abs/1712.07900

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