Abstract: We considered a class of discrete quasi-periodic Schrodinger operators with some Gevrey potential energy, in which the potential energy could be written as a large valued analytical function having a Gevrey small perturbation on the one-dimensional torus. By using large deviation theorem and semi-algebraic theory, we proved that the operator satisfied the non-perturbative Anderson localization for any fixed phase and almost all frequencies under large coefficients.

Key words: quasi-periodic Schrodinger operator, Gevrey , perturbation potential energy, large coupling coefficient, non-perturbative Anderson localization