On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential

Abstract

It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2\cos\left(\binom{n}{2}\omega +ny+x\right)$ with $\omega$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $\omega$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically verify the two conditions among these that are amenable to computer calculations.