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: vn = 2 cos n 2 ω + ny + x with ω an irrational number. Recently, Han, Schlag, and the second author derived a finitesize criterion in the case when ω 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.