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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: May 10, 2001
MathSciNet review: 1855620
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Abstract | References | Similar Articles | Additional Information

Abstract:

For the general one dimensional Schrödinger operator $-\frac{d^{2}}{dx^{2}}+q\left(x\right)$ with real $q\in L_{1} \left(\mathbb{R}\right) $ 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 $q$. These conditions are expressed in terms of the Fourier transform of some functions related to $q$. In particular, under the usual conditions $q^{\left( N\right) }\in L_{1}\left(\mathbb{R}\right), N\in\mathbb{N} _{0},$ 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: http://dx.doi.org/10.1090/S0002-9939-01-06014-2
PII: S 0002-9939(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