  In this paper, we consider some quasi-periodic Gevrey Jaocbi operators, which
can be defined by an analytic operators with a Gevrey small perturbation
potential, with the weak Diophantine frequency. We prove that if the Lyapunov
exponent of the original analytic operator is positive and the perturbation is
small enough, then the non-perturbative positivity and joint continuity of the
Lyapunov exponent, Anderson Localization and homogeneous spectrum hold for this
perturbative Gevrey operators.
