Search for: All records

We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $$2\lambda \cos\2π(j^2\omega+jy+x)$$. 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.