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

Abstract

For the general one-dimensional Schrödinger operator −d2/dx2+q(x) with real q ∈ L1(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) ∈ L1(x0, x0+δ), for N ∈ ℕ0, allows one to derive the (N+1) term for almost all x ∈ [x0, x0+δ), thereby refining a relevant result by Danielyan, Levitan and Simon.