WKB and spectral analysis of one-dimensional Schrodinger operators with slowly varying potentials

Consider a Schrodinger operator on L-2 Of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a fi nite sum of terms, each of which has a derivative of some order in L-1 + L- p for some exponent p < 2, then an essential support of the the absolutely continuous spectrum equals R+ Almost every generalized eigenfunction is bou nded, and satisfies certain WKB-type asymptotics at infinity. If moreover t hese derivatives belong to L-p with respect to a weight \x \ (gamma) with g amma > 0, then the Hausdorff dimension of the singular component of the spe ctral measure is strictly less than one.