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

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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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
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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. Yu. M. Shirokov, “Strongly singular potentials in three-dimensional quantum mechanics”, Theoret. and Math. Phys., 42:1 (1980), 28–31
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
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
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
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
6. Yu. G. Shondin, “Three-body problems with $\delta$-functional potentials”, Theoret. and Math. Phys., 51:2 (1982), 434–441
7. G. K. Tolokonnikov, “Shirokov algebras. I”, Theoret. and Math. Phys., 51:3 (1982), 554–561
8. I. S. Tsirova, “Singular potentials in a problem with noncentral interaction”, Theoret. and Math. Phys., 51:3 (1982), 561–563
9. V. A. Smirnov, “Associative algebra of functionals containing $\delta(x)$ and $r^n$”, Theoret. and Math. Phys., 52:2 (1982), 832–835
10. G. K. Tolokonnikov, “Differential rings used in Shirokov algebras”, Theoret. and Math. Phys., 53:1 (1982), 952–954
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
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
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
14. Theoret. and Math. Phys., 92:3 (1992), 1032–1037
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