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 TMF, 1979, Volume 39, Number 3, Pages 291–301 (Mi tmf2842)

Algebra of one-dimensional generalized functions

Yu. M. Shirokov

Abstract: An associative algebra $\mathscr{A}$, equipped with involution and differentiation, is constructed for generalized functions of one variable that at one fixed point can have singularities like the delta function and its derivatives and also finite discontinuities for the function and all its derivatives. The elements of $\mathscr{A}$ together with the differentiation operator form the algebra of local observables for a quantum theory with indefinite metric and state vectors that are also generalized functions. By going over to a smaller space, one can obtain quantum models with positive metric and with strongly singular concentrated potentials.

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English version:
Theoretical and Mathematical Physics, 1979, 39:3, 471–477

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Citation: Yu. M. Shirokov, “Algebra of one-dimensional generalized functions”, TMF, 39:3 (1979), 291–301; Theoret. and Math. Phys., 39:3 (1979), 471–477

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, “Algebra of three-dimensional generalized functions”, Theoret. and Math. Phys., 40:3 (1979), 790–794
2. Yu. M. Shirokov, “Strongly singular potentials in one-dimensional quantum mechanics”, Theoret. and Math. Phys., 41:2 (1979), 1031–1038
3. Yu. M. Shirokov, “Strongly singular potentials in three-dimensional quantum mechanics”, Theoret. and Math. Phys., 42:1 (1980), 28–31
4. Yu. M. Shirokov, “Representation of free solutions for Schrödinger equations with strongly singular concentrated potentials”, Theoret. and Math. Phys., 46:3 (1981), 191–196
5. 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
6. 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
7. O. G. Goryaga, Yu. M. Shirokov, “Energy levels of an oscillator with singular concentrated potential”, Theoret. and Math. Phys., 46:3 (1981), 210–212
8. G. K. Tolokonnikov, “Shirokov algebras. I”, Theoret. and Math. Phys., 51:3 (1982), 554–561
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. V. M. Shelkovich, “Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations”, Math. Notes, 57:5 (1995), 536–549
12. V. G. Danilov, V. P. Maslov, V. M. Shelkovich, “Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems”, Theoret. and Math. Phys., 114:1 (1998), 1–42
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