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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.

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Journal of Spectral Theory

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