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

Consider a Schrödinger operator on L2 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 L1 + Lp 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 Lp with respect to a weight |x|γ with γ > 0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.

Duke Scholars

Author Alexander A. Kiselev Mathematics