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

Published online by Cambridge University Press:  18 January 2002

ALEXEI RYBKIN
ALEXEI RYBKIN
Affiliation:
Department of Mathematical Sciences, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775; ffavr@uaf.edu
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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.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 1 , January 2002 , pp. 61 - 72
Copyright
© The London Mathematical Society 2002

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