We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.
Publication Source (Journal or Book title)
Ergodic Theory and Dynamical Systems
Han, R., Lemm, M., & Schlag, W. (2019). Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base. Ergodic Theory and Dynamical Systems, 40 (10), 2788-2853. https://doi.org/10.1017/etds.2019.19