On localization and the spectrum of multi-frequency quasi-periodic operators

We study multi-frequency quasi-periodic Schr\"odinger operators on $\mathbb{Z}$ in the regime of positive Lyapunov exponent and for general analytic potentials. Combining Bourgain's semi-algebraic elimination of multiple resonances with the method of elimination of double resonances via resultants, we establish exponential finite-volume localization as well as the separation between the eigenvalues. In a follow-up paper we develop the method further to show that for potentials given by large generic trigonometric polynomials the spectrum consists of a single interval, as conjectured by Chulaevski and Sinai.