Title: Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle Show Chinese title

Title: 魏尔和与斜移薛定谔cocycle的李雅普诺夫指数

Abstract: We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $2\lambda\cos\left(2\pi \left(\binom{j}{2} \omega+jy+x\right)\right)$. 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. Show Chinese abstract

Abstract: 我们研究具有偏移势的二维离散薛定谔算子$2\lambda\cos\left(2\pi \left(\binom{j}{2} \omega+jy+x\right)\right)$。 此势被长期猜想其行为类似于随机势，即期望对于任意小的耦合常数$\lambda>0$，它会产生安德森局域化。 在本文中，我们引入一种新的微扰方法来研究小$\lambda$时的零能李雅普诺夫指数$L(\lambda)$。 我们的主要结果表明，在微扰理论的二阶近似下，$L(\lambda)$的自然上界与$L(\lambda)$为正且满足通常的 Figotin-Pastur 类渐进行为$L(\lambda)\sim C\lambda^2$当$\lambda\to 0$时完全一致。 在 Almost-Mathieu 模型中，类似的量表现得完全不同，其零能李雅普诺夫指数对于$\lambda<1$为零。 主要的技术工作在于建立我们在微扰级数中出现的指数和（二次 Weyl 和）的良好下界。