Absolutely continuous spectrum of multifrequency quasiperiodic Schrödinger operator

Abstract In this paper, we prove that for any d-frequency analytic quasiperiodic Schrodinger operator, if the frequency is weak Liouvillean, and the potential is small enough, then the corresponding operator has absolutely continuous spectrum. Moreover, in the case d = 2 , we even establish the existence of ac spectrum under small potential and some super-Liouvillean frequency, and this result is optimal due to a recent counterexample of Avila and Jitomirskaya [6] .