Anderson localization for the discrete one-dimensional quasi-periodic Schroedinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = \lambda v (x + n \omega). We assume that the frequency \omega satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove - in the perturbative regime - that for large disorder \lambda and for most frequencies \omega the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.