Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems

In this paper, we prove that a quasi-periodic linear differential equation in sl(2,ℝ) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number and give some applications for Schrödinger operators. Our proof is a generalized KAM type iteration adapted to all irrational α.