A metal-insulator transition for the almost Mathieu model

We study the spectrum of the almost Mathieu hamiltonian: $$\left( {H_x \psi } \right)\left( n \right) = \psi \left( {n + 1} \right) + \psi \left( {n - 1} \right) + 2\mu \cos \left( {x - n\theta } \right)\psi \left( n \right),n \in \mathbb{Z}$$ where ϑ is an irrational number andx is in the circle\(\mathbb{T}\). For a small enough coupling constant μ and anyx there is a closed energy set of non-zero measure in the absolutely continuous spectrum ofH. For sufficiently high μ and almost allx we prove the existence of a set of eigenvalues whose closure has positive measure. The two results are obtained for a subset of irrational numbers ϑ of full Lebesgue measure.