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.

中文翻译：

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具有斜移势的薛定谔算子的有限大小李雅普诺夫指数

众所周知，当受到任意弱的随机势时，一维量子粒子是局域的。据推测，对于由非线性偏移动力学产生的任意弱势也会发生定位：$v_n=2\cos\left(\binom{n}{2}\omega +ny+x\right)$ 与 $ \omega$ 一个无理数。最近，Han、Schlag 和第二作者在 $\omega$ 是黄金平均值的情况下导出了一个有限大小的标准，它允许从在固定条件下施加的三个条件导出无限体积 Lyapunov 指数的正性，有限尺度。在这里，我们通过数值验证了其中适合计算机计算的两个条件。