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

TL;DR

This paper proves that for certain Schrödinger equations with analytic potentials, the Lyapunov exponent is non-perturbatively positive, with applications to skew-shift dynamics showing localization and spectral properties.

Contribution

It establishes non-perturbative positivity of Lyapunov exponents for a class of Schrödinger equations, including skew-shift cases, with implications for spectral theory.

Findings

Lyapunov exponent approximates the logarithm of the coupling number for large couplings.

Lyapunov exponent is weak H"older continuous in the skew-shift case.

Spectrum exhibits Anderson Localization and contains large intervals.

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.