Effective multi-scale approach to the Schr\"odinger cocycle over a skew shift base

TL;DR

This paper establishes a method to prove the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew shift base with a cosine potential, using finite-volume conditions and large deviation estimates.

Contribution

It introduces a multi-scale approach that makes existing techniques for Schrödinger operators with deterministic potentials effective for this setting.

Findings

Finite-volume conditions imply infinite-volume Lyapunov exponent positivity.

The approach is effective for coupling below the threshold of 1.

Demonstrates the applicability of large deviation estimates and the avalanche principle.

Abstract

We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schr\"odinger 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 Schr\"odinger operators with deterministic potentials, based on large deviation estimates and the avalanche principle, effective.