Almost Periodic Schrδdinger Operators III . The Absolutely Continuous Spectrum in One Dimension

We discuss the absolutely continuous spectrum of H = — d 2 /dx 2 + V(x) with F almost periodic and its discrete analog (hu)(n) = u(n +1) + u(n — 1) + V(ri)u(ri). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Fin the hull and a.e. E in A, H and h have continuum eigenfunctions, u 9 with \u\ almost periodic. In the discrete case, we prove that |^4|^4 with equality only if V= const. If k is the integrated density of states, we prove that on A, 2kdk/dE^π~ 2 in the continuum case and that 2πsmπkdk/dE^.l in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.