On Localization and the Spectrum of Multi-frequency Quasi-periodic Operators

We study multi-frequency quasi-periodic Schrödinger operators on Z in the regime of positive Lyapunov exponent and for general analytic potentials. Combining Bourgain’s semi-algebraic elimination of multiple resonances (Bourgain: Geom. Funct. Anal. 17 , 682–706, 2007) with the method of elimination of double resonances from Avila and Jitomirskaya (Ann. Math. (2) 173 , 337–475, 2011), we establish exponential finite-volume localization as well as the separation between the eigenvalues. In a follow-up paper (Goldstein et al.: Invent. Math. 217 , 603–701, 2019) 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 (Commun. Math. Phys. 125 , 91–112, 1989).