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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: with an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when 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.
Tobias Kippenberg, Guanhao Huang, Alberto Beccari, Amirali Arabmoheghi, Nils Johan Engelsen, Sergey Fedorov