Anderson Transitions for a Family of Almost Periodic Schrödinger Equations in the Adiabatic Case

This work is devoted to the study of a family of almost periodic one-dimensional Schr?dinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit, we prove the existence of an asymptotically sharp Anderson transition in the low energy region. More explicitly, we prove the existence of energy intervals containing only singular spectrum, and of other energy intervals containing absolutely continuous spectrum; the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions. The analysis may be viewed as utilizing a complex WKB method for adiabatic perturbations of periodic Schr?dinger equations. The transition energies are interpreted in terms of phase space tunneling. Received: 2 July 2001 / Accepted: 13 November 2001