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We consider perturbations of quasi-periodic Schr\"odinger operators on theinteger lattice with analytic sampling functions by decaying potentials andseek decay conditions under which various spectral properties are preserved. Inthe (almost) reducibility regime we prove that for perturbations with finitefirst moment, the essential spectrum remains purely absolutely continuous andthe newly created discrete spectrum must be finite in each gap of theunperturbed spectrum. We also prove that for fixed phase, Anderson localizationoccurring for almost all frequencies in the regime of positive Lyapunovexponents is preserved under exponentially decaying perturbations.

Stability of Spectral Types of Quasi-Periodic Schr\"odinger Operators With Respect to Perturbations by Decaying Potentials