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arXiv · 1904.08871

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

Paul Michael Kielstra·Marius Lemm
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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: $v_n=2\cos\left(\binom{n}{2}ω+ny+x\right)$ with $ω$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size 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.

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Paul Michael Kielstra, Marius Lemm. 2019-04-18. On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential. https://arxiv.org/abs/1904.08871

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