On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

Author:
Alexei Rybkin

Journal:
Proc. Amer. Math. Soc. **130** (2002), 59-67

MSC (2000):
Primary 34E05, 34L25; Secondary 34L40

DOI:
https://doi.org/10.1090/S0002-9939-01-06014-2

Published electronically:
May 10, 2001

MathSciNet review:
1855620

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Abstract | References | Similar Articles | Additional Information

For the general one dimensional Schrödinger operator with real we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness of . These conditions are expressed in terms of the Fourier transform of some functions related to . In particular, under the usual conditions we derive up to two extra terms in the asymptotic expansion of the Jost solution and for the transmission coefficient we derive twice as many terms. Our main results are complete.

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Additional Information

**Alexei Rybkin**

Affiliation:
Department of Mathematical Sciences, University of Alaska–Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775

Email:
ffavr@uaf.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06014-2

Keywords:
Schr\"{o}dinger operator,
asymptotic expansions,
Jost solution,
scattering matrix.

Received by editor(s):
April 18, 2000

Received by editor(s) in revised form:
May 15, 2000

Published electronically:
May 10, 2001

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2001
American Mathematical Society