Positive Lyapunov exponents and localization bounds for strongly mixing potentials

Abstract

For a one-dimensional discrete Schr\"odinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time.