Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle
We study the one-dimensional discrete Schrödinger operator with the skew-shift potential 2λ cos (2π((j/2)ω + jy C 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 λ > 0. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent L(λ) at small λ. Our main results establish that, to second order in perturbation theory, a natural upper bound on L(λ) is fully consistent with L(λ) being positive and satisfying the usual Figotin–Pastur type asymptotics L.λ/ ~ C λ2 as λ → 0. The analogous quantity behaves completely differently in the almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for λ < 1. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.
Publication Source (Journal or Book title)
Journal of Spectral Theory
Han, R., Lemm, M., & Schlag, W. (2021). Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle. Journal of Spectral Theory, 10 (4), 1139-1172. https://doi.org/10.4171/JST/323