Anderson localization for Schrodinger operators on Z with strongly mixing potentials

In this paper we show that for a.e. x epsilon (0, 2 pi) the operators defin ed on e(2)(Z(+)) as (H(x)psi)(n) = psi (n+1) + psi (n-1) + lambda cos(2(n) x)psi (n) for n grea ter than or equal to 0 and with Dirichlet condition psi -1 = 0, have pure point spectrum in [-2 delta, -delta] boolean OR [delta, 2 - delta] with exponentially decaying ei genfunctions where delta > 0 and 0 < \<lambda>\ < <lambda>(0)(delta) are sm all. As it is a simple consequence of known techniques that for small lambd a one has [-2 + delta, 2 - delta] subset of spectrum (H(x)) for a.e. x epsi lon (0, 2 pi), we thus established Anderson localization on the spectrum up to the edges and the center. More general potentials than cosine can be tr eated, but only those energies with nonzero spectral density are allowed. F inally, we prove the same result for operators on the whole line Z with pot ential v(n) (x) = lambda F(A(n)x), where A : T-2 --> T-2 is a hyperbolic to ral automorphism, F epsilon C-1(T-2), integral F = 0, and lambda small. The basis for our analysis is an asymptotic formula for the Lyapunov exponent for lambda --> 0 by Figotin-Pastur, and generalized by Chulaevski-Spencer. We combine this asymptotic expansion with certain martingale large deviatio n estimates in order to apply the methods developed by Bourgain and Goldste in in the quasi-periodic case.