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Reflectionless Potentials and Point Interactions in Pontryagin Spaces

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Abstract

The problem of constructing generalized point interactions of the second derivative operator in \(L^{2}(\mathbb{R})\) leading to the same scattering data as for reflectionless potentials is considered. It is proved that this problem has a solution only if extensions in Pontryagin spaces are involved. The solution of the inverse scattering problem is not unique, this is illustrated by considering the scattering data for soliton of the Korteweg-de Vries equation.

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Authors and Affiliations

  1. Department of Mathematics, Lund Institute of Technology, Box 118, 221 00, Lund, Sweden

    Pavel Kurasov

  2. Department of Physics, St. Petersburg University, 198504, St. Petersburg, Russia

    Pavel Kurasov

  3. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040, Wien, Austria

    Annemarie Luger

Authors
  1. Pavel Kurasov
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  2. Annemarie Luger
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Correspondence to Pavel Kurasov.

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MSC: 34B25, 47E05, 47B50, 81U40

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Kurasov, P., Luger, A. Reflectionless Potentials and Point Interactions in Pontryagin Spaces. Lett Math Phys 73, 109–122 (2005). https://doi.org/10.1007/s11005-005-0002-1

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  • DOI: https://doi.org/10.1007/s11005-005-0002-1

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