We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2\lambda\cos\big(2\pi \big(\binom{j}{2} \omega + jy + x\big)\big)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\lambda > 0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(\lambda)$ at small $\lambda$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(\lambda)$ is fully consistent with $L(\lambda)$ being positive and satisfying the usual Figotin–Pastur type asymptotics $L(\lambda)\sim C\lambda^2$ as $\lambda\to 0$. The analogous quantity behaves completely differently in the almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $\lambda < 1$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.

中文翻译：

---

偏移Schrödinger联合循环的Weyl sum和Lyapunov指数

我们研究具有偏斜移动势$ 2 \ lambda \ cos \ big（2 \ pi \ big（\ binom {j} {2} \ omega + jy + x \ big）\ big）$的一维离散Schrödinger算子。长期以来，人们一直猜想这种势能会表现得像一个随机势，即，对于任意小的耦合常数$ \ lambda> 0 $，它都会产生安德森局部化。在本文中，我们介绍了一种新颖的微扰方法，用于研究零能量Lyapunov指数$ L（\ lambda）$在小\\ lambda $的情况。我们的主要结果表明，对于扰动理论中的第二阶，$ L（\ lambda）$的自然上限与$ L（\ lambda）$为正并满足通常的Figotin-Pastur型渐近线$ L（ \ lambda）\ sim C \ lambda ^ 2 $作为$ \ lambda \至0 $。在几乎Mathieu模型中，类似量的行为完全不同，其零能量Lyapunov指数在$ \ lambda <1 $时消失。主要技术工作在于为出现在我们的扰动序列中的指数和（二次Weyl和）建立良好的下界。