| In the first chapter, the eigenvalue problem for a periodic Schrodinger operator, Lf = (-d('2)/dx('2) + v)f = Ef, is viewed as a two-dimensional Hamiltonian system which is integrable in the sense of Arnold and Liouville. With the aid of the Floquet-Bloch theory, it is shown that such a system is conjugate to two harmonic oscillators with frequencies (alpha) and (omega), being the rotation number for L and 2(pi)/(omega) the period of the potential v.; This picture is generalized in the second chapter, to quasi- periodic Schrodinger operators, L(,(epsilon)), with highly irrational frequencies ((omega)(,1), ..., (omega)(,d)), which are a small perturbation of periodic operators. When the parameter E belongs to a certain (explicitly constructed) large Cantor set E, the eigenvalue problem for L(,(epsilon)) is embedded, via a KAM method, in a system of d + 1 harmonic oscillators with frequencies ((alpha), (omega)(,1), ..., (omega)(,d)), (alpha) being the (Johnson-Moser) rotation number for L(,(epsilon)).; A by-product of the above is that, on E, all the eigensolutions of L(,(epsilon)) are quasi-periodic with frequencies ((alpha), (omega)(,1), ..., (omega)(,d)) and depend smoothly on E (in the Whitney sense).; In the last chapter, the absolutely continuous spectrum (sigma)(,ac) of a general quasi-periodic Schrodinger operator is considered. The Radon-Nikodym derivatives (with respect to Lebesgue measure) of the spectral measures are computed in terms of special independent eigensolutions existing for almost every E in (sigma)(,ac). Such eigensolutions can be replaced, in the above Radon-Nikodym derivatives, by Bloch waves whenever these exist (as in the case treated in Chapter 2).; Finally, it is shown that weak Bloch waves always exist for almost every E in (sigma)(,ac) and the question of the existence of genuine Bloch waves is turned into a regularity problem for a certain nonlinear partial differential equation on a d-dimensional torus. |
