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Mathematics of the USSR-Sbornik, 1989, Volume 64, Issue 1, Pages 161–175
DOI: https://doi.org/10.1070/SM1989v064n01ABEH003300
(Mi sm1734)
 

This article is cited in 29 scientific papers (total in 29 papers)

Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schrödinger operator with pointwise potential

B. S. Pavlov
References:
Abstract: The purpose of this article is to describe the formulation of a selfadjoint spectral problem with boundary conditions on a sufficiently thin manifold. Namely, let $\mathscr L$ be a selfadjoint operator in $L_2(\mathbf R^n)$, let $L$ be a smooth manifold, let $\mathscr L_0$ be the restriction of $\mathscr L$ to the lineal in $\mathscr D(\mathscr L_0)$ consisting of all functions which vanish in a neighborhood of $L$.
It is shown that the deficiency elements of this restriction can be represented as “tensor layers” with densities of a definite class of smoothness, concentrated on the “boundary” of $L$. If $L$ is sufficiently thin, there is only one family of deficiency elements, and it is analogous to the single-layer potentials. In this case, calculation of the boundary form and the description of the selfadjoint extensions appears to be quite simple. This case is studied in detail because the investigation of the simplest model of the three-particle problem of quantum mechanics reduces to it.
Bibliography: 16 titles.
Received: 14.05.1987
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1988, Volume 136(178), Number 2(6), Pages 163–177
Bibliographic databases:
UDC: 517.9
MSC: Primary 35J10; Secondary 35P20, 81F10
Language: English
Original paper language: Russian
Citation: B. S. Pavlov, “Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schrödinger operator with pointwise potential”, Mat. Sb. (N.S.), 136(178):2(6) (1988), 163–177; Math. USSR-Sb., 64:1 (1989), 161–175
Citation in format AMSBIB
\Bibitem{Pav88}
\by B.~S.~Pavlov
\paper Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schr\"odinger operator with pointwise potential
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 136(178)
\issue 2(6)
\pages 163--177
\mathnet{http://mi.mathnet.ru/sm1734}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=954922}
\zmath{https://zbmath.org/?q=an:0687.35064}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 1
\pages 161--175
\crossref{https://doi.org/10.1070/SM1989v064n01ABEH003300}
Linking options:
  • https://www.mathnet.ru/eng/sm1734
  • https://doi.org/10.1070/SM1989v064n01ABEH003300
  • https://www.mathnet.ru/eng/sm/v178/i2/p163
  • This publication is cited in the following 29 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:376
    Russian version PDF:116
    English version PDF:13
    References:54
     
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