Metal-insulator transition for the almost Mathieu operator

Abstract

We prove that for Diophantine \om and almost every þ, the almost Mathieu operator, (H_{ω,λ,θ}Ψ)(n)=Ψ(n+1) + Ψ(n-1) + λ\cos 2π(ωn +θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ< 2. This completes the proof of (a correct version of) the Aubry-André conjecture.