SOME NEW AND OLD ASYMPTOTIC REPRESENTATIONS OF THE JOST SOLUTION AND THE WEYL <i>m</i>-FUNCTION FOR SCHRÖDINGER OPERATORS ON THE LINE

Abstract

For the general one-dimensional Schrödinger operator − d 2 / dx 2 + q ( x ) with real q ∈ L 1 (ℝ), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m -function for locally summable q . This representation is then applied to smooth potentials q to obtain Weyl m -function power asymptotics. The condition q ( N ) ∈ L 1 ( x 0 , x 0 +δ), for N ∈ ℕ 0 , allows one to derive the ( N +1) term for almost all x ∈ [ x 0 , x 0 +δ), thereby refining a relevant result by Danielyan, Levitan and Simon.