$L^2$-reducibility and localization for quasiperiodic operators

Abstract

We give a simple argument that if a quasiperiodic multi-frequency Schr\"odinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then the spectrum of the dual operator is purely point for Lebesgue almost all values of the ergodic parameter $\theta$. The result holds in the $L^2$ setting provided, in addition, that the conjugation preserves the fibered rotation number. Corollaries include localization for (long-range) 1D analytic potentials with dual ac spectrum and Diophantine frequency as well as a new result on multidimensional localization.