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A convexity theorem in the scattering theory for the Dirac operator


Author: K. L. Vaninsky
Journal: Trans. Amer. Math. Soc. 350 (1998), 1895-1911
MSC (1991): Primary 34L05, 34L25, 34L40
DOI: https://doi.org/10.1090/S0002-9947-98-02150-3
MathSciNet review: 1467476
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Abstract: The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrödinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrödinger equation.


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Additional Information

K. L. Vaninsky
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: vaninsky@math.ksu.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02150-3
Keywords: Scattering theory, convexity theorem, nonlinear Schr\"{o}dinger
Received by editor(s): November 9, 1995
Received by editor(s) in revised form: June 21, 1996
Additional Notes: The author would like to thank the Institut des Hautes Études Scientifiques, where the paper was completed, for hospitality. The work is partially supported by NSF grant DMS-9501002
Article copyright: © Copyright 1998 American Mathematical Society