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

摘要

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 finite 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 bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to L-p with respect to a weight x (gamma) with gamma > 0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one. [References: 32]