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TMF, 1979, Volume 40, Number 3, Pages 348–354 (Mi tmf2912)  

This article is cited in 14 scientific papers (total in 14 papers)

Algebra of three-dimensional generalized functions

Yu. M. Shirokov

Abstract: The method proposed by the author in an earlier paper [1] is used to construct the associative algebra $\mathscr{A}$$(3)$, which is equipped with involution and differentiation, for generalized functions of three variables that at a fixed point can have singularities of the type $\delta(\mathbf{r})$, $r^{-1}$, $r^{-2}$ and their derivatives. In complete analogy with the one-dimensional algebra of [1], the elements of the algebra $\mathscr{A}$$(3)$ form in conjunction with the differentiation operator an algebra of local operators of quantum theory with indefinite metric and with state vectors that are also generalized functions. It is noted that one can go over to smaller spaces of state vectors and obtain three-dimensional Schrödinger equations with strongly singular potentials and positive metric.

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English version:
Theoretical and Mathematical Physics, 1979, 40:3, 790–794

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Received: 16.04.1979

Citation: Yu. M. Shirokov, “Algebra of three-dimensional generalized functions”, TMF, 40:3 (1979), 348–354; Theoret. and Math. Phys., 40:3 (1979), 790–794

Citation in format AMSBIB
\by Yu.~M.~Shirokov
\paper Algebra of three-dimensional generalized functions
\jour TMF
\yr 1979
\vol 40
\issue 3
\pages 348--354
\jour Theoret. and Math. Phys.
\yr 1979
\vol 40
\issue 3
\pages 790--794

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    This publication is cited in the following articles:
    1. Yu. M. Shirokov, “Strongly singular potentials in three-dimensional quantum mechanics”, Theoret. and Math. Phys., 42:1 (1980), 28–31  mathnet  crossref  mathscinet  isi
    2. Yu. M. Shirokov, “Representation of free solutions for Schrödinger equations with strongly singular concentrated potentials”, Theoret. and Math. Phys., 46:3 (1981), 191–196  mathnet  crossref  mathscinet  zmath  isi
    3. G. K. Tolokonnikov, Yu. M. Shirokov, “Associative algebra of generalized functions closed with respect to differentiation and integration”, Theoret. and Math. Phys., 46:3 (1981), 200–203  mathnet  crossref  mathscinet  zmath  isi
    4. I. S. Tsirova, Yu. M. Shirokov, “Quantum delta-like potential acting in the $P$ state”, Theoret. and Math. Phys., 46:3 (1981), 203–206  mathnet  crossref  mathscinet  zmath  isi
    5. S. V. Talalov, Yu. M. Shirokov, “Interaction of a charged particle with an external electromagnetic field in the presence of a strongly singular potential”, Theoret. and Math. Phys., 46:3 (1981), 207–210  mathnet  crossref  mathscinet  isi
    6. Yu. G. Shondin, “Three-body problems with $\delta$-functional potentials”, Theoret. and Math. Phys., 51:2 (1982), 434–441  mathnet  crossref  mathscinet  isi
    7. G. K. Tolokonnikov, “Shirokov algebras. I”, Theoret. and Math. Phys., 51:3 (1982), 554–561  mathnet  crossref  mathscinet  zmath  isi  elib
    8. I. S. Tsirova, “Singular potentials in a problem with noncentral interaction”, Theoret. and Math. Phys., 51:3 (1982), 561–563  mathnet  crossref  isi
    9. V. A. Smirnov, “Associative algebra of functionals containing $\delta(x)$ and $r^n$”, Theoret. and Math. Phys., 52:2 (1982), 832–835  mathnet  crossref  mathscinet  zmath  isi
    10. G. K. Tolokonnikov, “Differential rings used in Shirokov algebras”, Theoret. and Math. Phys., 53:1 (1982), 952–954  mathnet  crossref  mathscinet  zmath  isi
    11. Yu. G. Shondin, “Generalized pointlike interactions in $R_3$ and related models with rational $S$-matrix”, Theoret. and Math. Phys., 64:3 (1985), 937–944  mathnet  crossref  mathscinet  isi
    12. B. S. Pavlov, “Electron in a homogeneous crystal of point atoms with internal structure. I”, Theoret. and Math. Phys., 72:3 (1987), 964–972  mathnet  crossref  mathscinet  isi
    13. Yu. G. Shondin, “Quantum-mechanical models in $R_n$ associated with extensions of the energy operator in a Pontryagin space”, Theoret. and Math. Phys., 74:3 (1988), 220–230  mathnet  crossref  mathscinet  zmath  isi
    14. Theoret. and Math. Phys., 92:3 (1992), 1032–1037  mathnet  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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