On a complete analysis of highenergy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials
Author:
Alexei Rybkin
Journal:
Proc. Amer. Math. Soc. 130 (2002), 5967
MSC (2000):
Primary 34E05, 34L25; Secondary 34L40
Published electronically:
May 10, 2001
MathSciNet review:
1855620
Fulltext PDF Free Access
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Abstract: 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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 5.
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 A.V. Rybkin, ``A Weyl function approach to trace formulas for varies Schrödinger operators", in preparation.
 15.
 A.V. Rybkin, ``Some new and old asymptotic representations of the Jost solution and Weyl function for Schrödinger operators on the line", to appear in Bulletin of LMS.
 16.
 B. Simon, ``A new approach to inverse spectral theory, I. Fundamental formalism", Annals of Math. 150 (1999), 10291057. CMP 2000:08
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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:
http://dx.doi.org/10.1090/S0002993901060142
PII:
S 00029939(01)060142
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
