WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Abstract:

Consider a Schrödinger 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|^γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.