Weyl sums and the Lyapunov exponent for the skew-shift Schr\"odinger cocycle

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.