Effective multi-scale approach to the Schrödinger cocycle over a skew shift base

Abstract

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.

References in corpus (2)

• The distribution of spacings between the fractional parts of $n^2\alpha$
• Pair Correlation for Fractional Parts of $\alpha n^2$

Cited by in corpus (5)

• Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities
• Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle
• Global eigenvalue distribution of matrices defined by the skew-shift
• On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential
• A Local Law for Singular Values from Diophantine Equations