Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle

Abstract

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.