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Zap. Nauchn. Sem. POMI, 2015, Volume 433, Pages 78–110 (Mi znsl6128)  

This article is cited in 4 scientific papers (total in 4 papers)

Extensions of the quadratic form of the transverse Laplace operator

T. A. Bolokhov

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We review the quadratic form of the Laplace operator in spehrical coordinates which acts on the transverse components of vector functions on the $3$-dimensional space. Operators, acting on the parametrizing functions of one of the transverse components with angular momentum 1 and 2, appear to be fourth order symmetric differential operators with deficiency indices (1,1). We develop self-adjoint extensions of these operators and propose correspondent extensions for the initial quadratic form. Eigenfuctions of the extensions in question represent a stable soliton-like solutions of the physical system with the quadratic form being a potential energy.

Key words and phrases: self-adjoint extensions of symmetric operators, quadratic forms, Laplace operator, transverse subspace.

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English version:
Journal of Mathematical Sciences (New York), 2016, 213:5, 671–693

Bibliographic databases:

UDC: 517.9
Received: 11.03.2015

Citation: T. A. Bolokhov, “Extensions of the quadratic form of the transverse Laplace operator”, Questions of quantum field theory and statistical physics. Part 23, Zap. Nauchn. Sem. POMI, 433, POMI, St. Petersburg, 2015, 78–110; J. Math. Sci. (N. Y.), 213:5 (2016), 671–693

Citation in format AMSBIB
\Bibitem{Bol15}
\by T.~A.~Bolokhov
\paper Extensions of the quadratic form of the transverse Laplace operator
\inbook Questions of quantum field theory and statistical physics. Part~23
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 433
\pages 78--110
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6128}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3493681}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 213
\issue 5
\pages 671--693
\crossref{https://doi.org/10.1007/s10958-016-2731-3}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84957623938}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. T. A. Bolokhov, “Properties of the $l=1$ radial part of the Laplace operator in a special scalar product”, J. Math. Sci. (N. Y.), 215:5 (2016), 560–573  mathnet  crossref  mathscinet
    2. T. A. Bolokhov, “Properties of some extensions of the quadratic form of the vector Laplace operator”, J. Math. Sci. (N. Y.), 229:5 (2018), 487–496  mathnet  crossref  mathscinet
    3. T. A. Bolokhov, “Resolvents of selfadjoint extensions of the Laplace operator on the solenoidal subspace”, J. Math. Sci. (N. Y.), 243:6 (2019), 835–840  mathnet  crossref
    4. T. A. Bolokhov, “Scalar products for the regular analytic vectors of the Laplace operator in the solenoidal subspace”, J. Math. Sci. (N. Y.), 242:5 (2019), 642–650  mathnet  crossref
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