In the paper we consider the discrete quasi-periodic Schrödinger operator under a perturbation of the potential, and establish the Anderson localization of the perturbed operator at the fixed initial phase and almost every frequency, provided that the unperturbed potential is large and the perturbation is small. The conclusion can solve the nearly degenerate two-frequency problem that Bourgain addressed in his study. The proof relies on continuity properties of the Lyapunov exponent, the large deviation theorem, and the avalanche principle.
Zou et al. (Fri,) studied this question.