Almost periodic Schrödinger operators

We discuss the absolutely continuous spectrum ofH=?d ²/dx ²+V(x) withV almost periodic and its discrete analog (hu)(n)=u(n+1)+u(n?1)+V(n)u(n). 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.V in the hull and a.e.E inA, H andh have continuum eigenfunctions,u, with |u| almost periodic. In the discrete case, we prove that |A|≦4 with equality only ifV=const. Ifk is the integrated density of states, we prove thaton A, 2kdk/dE≧π^?2 in the continuum case and that 2π sinπkdk/dE≧1 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.