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 Mat. Sb. (N.S.), 1975, Volume 97(139), Number 1(5), Pages 77–93 (Mi msb3485)

Calculating losses in scattering problems

B. S. Pavlov

Abstract: In this article we solve the problem about the calculation of losses in a scattering problem with “Lax” and “non-Lax” channels. For the initial scattering matrix we consider the scattering matrix of the basic operator of the problem with respect to a simple unperturbed operator, which acts in a distinguished subspace (a Lax channel) that is the orthogonal sum of the incoming and outgoing subspaces. It turns out that this scattering matrix is nonunitary when the basic space contains other channels besides the distinguished one, including non-Lax channels. The concept of losses is connected with the fact that the scattering matrix is nonunitary. We calculate the losses by constructing in the orthogonal complement of a Lax channel a new selfadjoint operator, which with the original unperturbed operator forms a modified unperturbed operator. The latter has a unitary scattering matrix with respect to the basic operator of the problem. We explain the significance of the elements of the new scattering matrix that include the original matrix.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:1, 71–87

Bibliographic databases:

UDC: 513.88
MSC: Primary 47A40; Secondary 47B44, 81A48

Citation: B. S. Pavlov, “Calculating losses in scattering problems”, Mat. Sb. (N.S.), 97(139):1(5) (1975), 77–93; Math. USSR-Sb., 26:1 (1975), 71–87

Citation in format AMSBIB
\Bibitem{Pav75} \by B.~S.~Pavlov \paper Calculating losses in scattering problems \jour Mat. Sb. (N.S.) \yr 1975 \vol 97(139) \issue 1(5) \pages 77--93 \mathnet{http://mi.mathnet.ru/msb3485} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=385605} \zmath{https://zbmath.org/?q=an:0325.47007} \transl \jour Math. USSR-Sb. \yr 1975 \vol 26 \issue 1 \pages 71--87 \crossref{https://doi.org/10.1070/SM1975v026n01ABEH002470} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. V. Rybkin, “The spectral shift function for a dissipative and a selfadjoint operator, and trace formulas for resonances”, Math. USSR-Sb., 53:2 (1986), 421–431
2. Rybkin A., “The Trace Formula for Dissipative and Self-Adjoint Operators - Spectral Identities for Resonances”, no. 4, 1984, 97–99
3. Neidhardt H., “On the Inverse Problem of a Dissipative Scattering-Theory .3.”, Math. Nachr., 148 (1990), 229–242
4. A. V. Rybkin, “The spectral shift function, the characteristic function of a contraction, and a generalized integral”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 237–281
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