ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL HELGE KR ¨ UGER Abstract. I will prove that for most energies E and some frequencies α the Lyapunov exponent of the Schr¨ odinger operator with potential V (n)= 2λ cos(2παn 2 ) has the behavior L(E)  λ 2 as λ → 0. This improves upon earlier results by Bourgain. 1. Introduction In this paper, I wish to analyze the skew-shift Schr¨odinger operator at small coupling. That is for λ> 0 small and α irrational the potential (1.1) V α,λ (n)=2λ cos(2παn 2 ) and the associated Schr¨odinger operator H α,λ = Δ+ V α,λ . The Lyapunov exponent L α,λ (E) describes the maximal exponential growth of solutions of H α,λ u = Eu. We will give a definition in (2.4). See also [7] or [15] for background. It is known that for irrational α and λ> 1 large enough, we have L α,λ (E)  log λ. This can be shown by using Herman’s subharmonicity trick from [11], or under suitable additional assumptions on α or E by the more constructive methods of Bourgain [3], Bourgain, Goldstein, and Schlag [6], or myself [12]. My main goal is to improve the result of Bourgain from [1] on positivity of the Lyapunov exponent for this model for small λ> 0, certain irrational α, and most energies E. I will prove the following quantitative version of Bourgain’s result. Theorem 1.1. Given δ> 0 and τ> 0. There are λ 0 = λ 0 (δ, τ ) > 0 and γ 0 = γ 0 (δ, τ ) > 0. For each 0 <λ<λ 0 , there is α 0 (λ) > 0 such that for 0 <α<α 0 irrational, there exists a set E b = E b (λ, α) of measure (1.2) |E b |≤ τ such that for E ∈ ([−2+ δ, −δ] ∪ [δ, 2 − δ]) \E b (1.3) L α,λ (E) ≥ γ 0 λ 2 . The main improvement is the quantitative lower bound L α,λ (E)  λ 2 . This behavior is expected to hold for all irrational α, E ∈ (−2, 2) \{0}, and 0 <λ ≤ 1. However even positivity is unknown (see e.g. [4], [14]). In my earlier paper [12], I established a similar result for E ∈ (−2, 2) \{0}. However, I had to replace the Date : April 27, 2010. 2000 Mathematics Subject Classification. Primary 81Q10; Secondary 37D25. Key words and phrases. Lyapunov Exponents, Schr¨ odinger Operators. H. K. was supported by NSF grant DMS–0800100 and a Nettie S. Autrey Fellowship. 1