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On D. Ya. Petrina's works in contemporary mathematical physics

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Ukrainian Mathematical Journal Aims and scope

Abstract

This is a brief survey of the works of Prof. D. Ya. Petrina in various branches of contemporary mathematical physics.

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References

  1. D. Ya. Petrina, O. S. Parasyuk, and P. N. Tatsunyak, “Chellen-Lehmann theorem for quantum field theory in a space with indefinite metric,”Ukr. Mat. Zh.,10, No. 3, 344–346 (1958).

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  2. D. Ya. Petrina, “Dispersion relations for inelastic scattering in nonrelativistic approximation,”Ukr. Mat. Zh.,11, No. 3, 267–274 (1959).

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  3. D. Ya. Petrina, “A solution of the inverse diffraction problem,”Ukr. Mat. Zh.,12, No. 4, 476–479 (1960).

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  4. D. Ya. Petrina, “On the impossibility of a nonlocal field theory with a positive spectrum of the energy-momentum operator,”Ukr. Mat. Zh.,13, No. 4, 109–111 (1961).

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  5. D. Ya. Petrina, “Analytic properties of partial waves of the scattering amplitude in perturbation theory,”Dokl. Akad Nauk SSSR,144, No. 4, 755–758 (1962).

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  6. D. Ya. Petrina, “Analytic properties of the scattering amplitude for a potential on the first “nonphysical” sheet,”Zh. Éksp. Teor. Fiz.,44, Issue 1, 151–156 (1963).

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  7. D. Ya. Petrina, “Analytic properties of the contributions of Feynman diagrams,”Dokl. Akad Nauk SSSR,149, No. 4, 808–810 (1963).

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  8. D. Ya. Petrina, “Complex singular points of the contributions of Feynman diagrams and the continuity theorem,”Ukr. Mat. Zh.,16, No. 1, 31–40 (1964).

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  9. D. Ya. Petrina, “On the principle of maximal analyticity with respect to the complex orbital momentum,”Ukr. Mat. Zh.,16, No. 4, 502–512 (1964).

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  10. D. Ya. Petrina, “Mandelstam representation and the continuity theorem,”Zh. Éksp. Teor. Fiz.,46, Issue 2, 544–554 (1964).

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  11. D. Ya. Petrina, “A proof of the Mandelstam representation for a ladder diagram of the sixth order,”Zh. Éksp. Teor. Fiz.,47, Issue 8, 524–529 (1964).

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  12. D. Ya. Petrina, “Analytic properties of a class of functions defined by integrals over a manifold. I,”Ukr. Mat. Zh.,17, No. 5, 54–66 (1965).

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  13. D. Ya. Petrina, “Analytic properties of a class of functions defined by integrals over a manifold. II,”Ukr. Mat. Zh.,17, No. 6, 60–66 (1965).

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  14. D. Ya. Petrina, “On a holomorphic extension of the contributions of Feynman diagrams,”Dokl. Akad Nauk SSSR,168, No. 2, 308–309 (1966).

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  15. D. Ya. Petrina, “On the completeness of amplitudes of perturbation theory in the space of amplitudes,”Ukr. Mat. Zh.,19, No. 3, 62–78 (1967).

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  16. D. Ya. Petrina, “On the completeness of amplitudes of perturbation theory in the space of amplitudes,”Dokl. Akad Nauk SSSR,173, No. 2, 295–297 (1967).

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  17. D. Ya. Petrina, “On the summation of contributions of Feynman diagrams: the existence theorem,”Izv. Akad Nauk SSSR, Ser. Mat.,32, No. 5, 1052–1074 (1968).

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  18. D. Ya. Petrina,Mathematical Problems in the Theory of Scattering Matrices, Doctor's Degree Thesis (Physics and Mathematics), Kiev (1968).

  19. D. Ya. Petrina,Mathematical Problems in the Theory of Scattering Matrices, Author's Abstract of the Doctor's Degree Thesis (Physics and Mathematics), Kiev (1968).

  20. D. Ya. Petrina and S. S. Ivanov, “On equations appearing as a result of summation of perturbation series for scattering matrices,”Dokl. Akad Nauk SSSR,188, No. 4, 776–779 (1969).

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  21. D. Ya. Petrina, N. N. Bogolyubov, and B. I. Khatset, “Mathematical description of equilibrium states of classical systems on the basis of the formalism of canonical ensemble,”Teor. Mat. Fit.,1, No. 2, 251–274 (1969).

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  22. D. Ya. Petrina, “On Hamiltonians of quantum statistics and on the model Hamiltonian in the theory of superconductivity,”Teor. Mat. Fiz.,4, No. 3, 394–411 (1970).

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  23. D. Ya. Petrina and V. I. Skrypnyk, “Kirkwood-Salsburg equations for coefficient functions of the scattering matrix,”Teor. Mat. Fiz.,8, No. 3, 369–380 (1971).

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  24. D. Ya. Petrina and V. P. Yatsyshyn, “On the model Hamiltonian in the theory of superconductivity,”Teor. Mat. Fiz.,10, No. 2, 283–299 (1972).

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  25. D. Ya. Petrina, “On solutions of the Bogolyubov kinetic equations. Quantum statistics,”Teor. Mat. Fiz.,13, No. 3, 391–405 (1972).

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  26. D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov kinetic equations,”Tr. Mat. Inst. Akad. Nauk SSSR,136, 370–378 (1975).

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  27. D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “On equations for the coefficient functions of theS-matrix in quantum field theory,”Teor. Mat. Fiz.,19, No. 1, 37–46 (1974).

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  28. D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “S-matrix in constructive field theory,”Teor. Mat. Fiz.,23, No. 2, 160–177 (1975).

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  29. D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “S-matrix in constructive field theory,”Fiz. Element. Chastic Atom. Yadra,7, Issue 3, 647–686 (1976).

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  30. D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov equations,”Dokl. Akad Nauk SSSR,228, No. 3, 573–575 (1976).

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  31. D. Ya. Petrina and V. Z. Énol'skii,On Oscillations of One-Dimensional Systems [in Russian], Preprint, Institute of Theoretical Physics, Ukrainian Academy of Sciences, Kiev (1976).

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  32. D. Ya. Petrina and V. Z. Énol'skii, “On oscillations of one-dimensional systems,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 8, 756–760 (1976).

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  33. D. Ya. Petrina and N. N. Bogolyubov (Jr.), “On a class of model systems that admit a reduction of the degree of the Hamiltonian in the thermodynamic limit. I,”Teor. Mat. Fiz.,33, No. 2, 231–245 (1977).

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  34. D. Ya. Petrina and N. N. Bogolyubov (Jr.), “On a class of model systems that admit a reduction of the degree of the Hamiltonian in the thermodynamic limit. II,”Teor. Mat. Fiz.,37, No. 2, 246–257 (1978).

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  35. D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko,Equations for the Coefficient Functions of the Scattering Matrix [in Russian], Nauka, Moscow (1979).

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  36. D. Ya. Petrina, “Mathematical description of the evolution of infinite systems of classical statistical physics. Locally perturbed one-dimensional systems,”Teor. Mat. Fiz.,38, No. 2, 230–250 (1979).

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  37. D. Ya. Petrina and V. I. Gerasimenko, “Statistical mechanics of quantum-classical systems. Nonequilibrium systems,”Teor. Mat. Fiz.,42, No. 1, 88–100 (1980).

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  38. D. Ya. Petrina and A. L. Rebenko, “Projection-iterative method for the solution of equations of quantum field theory and its connection with renormalization theory. Equations of quantum field theory and ill-posed problems of mathematical physics,”Teor. Mat. Fiz.,42, No. 2, 167–183 (1980).

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  39. D. Ya. Petrina, V. I. Gerasimenko, A. I. Pilyavskii, and P. V. Malyshev, “On the process of inverse osmosis as a boundary-value problem in domains with complicated structure,”Dokl Akad Nauk Ukr. SSR, Ser. A., No. 9, 75–78 (1980).

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  40. D. Ya. Petrina and A. I. Pilyavskii,Potential of the Electrostatic Field of Charged Particles and a Dynamical Membrane [in Russian], Preprint 80-141P, Institute of Theoretical Physics, Ukrainian Academy of Sciences, Kiev (1980).

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  41. D. Ya. Petrina and A. I. Pilyavskii, “Potential of the electrostatic field of a system of charged particles and a dynamical membrane,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 7, 57–60 (1981).

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  42. D. Ya. Petrina, “Solution of a classical problem of electrostatics and the subtraction procedure,”Dokl. Akad Nauk SSSR,270, No. 1, 78–81 (1983).

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  43. D. Ya. Petrina and V. I. Gerasimenko, “Mathematical description of the evolution of states of infinite systems of classical statistical mechanics,”Usp. Mat. Nauk.,38, No. 5, 5–61 (1983).

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  44. D. Ya. Petrina, “Solution of a classical problem of electrostatics by using the subtraction procedure,”Zh. Vych. Mat. Mat. Fiz.,24, No. 5, 709–721 (1984).

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  45. D. Ya. Petrina,Quantum Field Theory [in Russian], Vyshcha Shkola, Kiev (1984).

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  46. D. Ya. Petrina and E. D. Belokolos, “On the relation between the methods of approximating Hamiltonian and finite-zone integration,”Dokl. Akad Nauk SSSR,275, No. 3, 580–582 (1984).

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  47. D. Ya. Petrina and E. D. Belokolos, “On the relation between the methods of approximating Hamiltonian and finite-zone integration,”Teor. Mat. Fiz.,58, No. 1, 61–71 (1984).

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  48. D. Ya. Petrina, “Distribution functions of systems of charged particles in spatially-inhomogeneous media,”Teor. Mat. Fiz.,60, No. 1, 104–116 (1985).

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  49. D. Ya. Petrina and V. I. Gerasimenko, “Evolution of states of infinite systems of classical statistical mechanics,”Sov. Sci. Rev., Ser. C,5, 1–51 (1985).

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  50. D. Ya. Petrina and A. I. Pilyavskii, “Problems of electrostatics in spatially-inhomogenecus media and the subtraction procedure,”Fiz. Mnogochastich. Sist., Issue7, 82–96 (1985).

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  51. D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic limit for nonequilibrium states of a three-dimensional system of hard spheres,”Teor. Mat. Fiz.,64, No. 1, 130–149 (1985).

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  52. D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic limit for nonequilibrium states of a three-dimensional system of hard spheres,”Dokl. Akad Nauk SSSR,282, No. 1, 130–136 (1985).

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  53. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev,Mathematical Foundations of Classical Statistical Mechanics [in Russian], Naukova Dumka, Kiev (1985).

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  54. D. Ya. Petrina, Yu. A. Mitropolskii, and V. G. Baryakhtar, “Creative contribution of Academician N. N. Bogolyubov to the development of mathematics, nonlinear mechanics, and theoretical physics,”Vestn. Akad. Nauk Ukr. SSR, No. 11, 9–21 (1985).

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  55. D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic and Boltzmann-Grad limits for nonequilibrium states of a system of hard spheres,” in:Problems of Modern Statistical Physics [in Russian], Naukova Dumka, Kiev (1985), pp. 228–337.

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  56. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, “Thermodynamic limit for solutions of the Bogolyubov equations,”Sov. Sci. Rev., Ser. C,7, 280–336 (1987).

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  57. D. Ya. Petrina, “Mathematical problems of the description of the evolution of states of infinite systems of statistical mechanics,” in:Proceedings of the Internat. Workshop “Selected Problems of Statistical Physics,” Vol. 2, World Scientific, Singapore (1987), p. 332.

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  58. D. Ya. Petrina and V. I. Gerasimenko, “Boltzmann-Grad limit for the states of an infinite system of hard spheres,”Dokl. Akad Nauk SSSR,297, No. 2, 336–340 (1987).

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  59. D. Ya. Petrina, A. N. Bogolyubov, and T. M. Urbanskii,Nikolai Mitrofanovich Krylov [in Russian], Naukova Dumka, Kiev (1987), pp. 157–169.

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  60. D. Ya. Petrina and P. V. Malyshev, “Thermodynamic limit for nonequilibrium distributions functions of three-dimensional classical systems of interacting particles,”Dokl. Akad Nauk SSSR,301, No. 3, 585–589 (1988).

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  61. D. Ya. Petrina and A. V. Mishchenko, “Exact solutions a class of Boltzmann equations,”Dokl. Akad Nauk SSSR,298, No. 2, 338–342 (1988).

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  62. D. Ya. Petrina and A. V. Mishchenko, “Linearization and exact solutions a class of Boltzmann equations,”Teor. Mat. Fiz.,77, No. 1, 135–153 (1988).

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  63. D. Ya. Petrina and V. I. Gerasimenko, “Boltzmann-Grad limit for equilibrium states,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 12, 17–19 (1988).

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  64. D. Ya. Petrina and V. I. Gerasimenko, “On the limiting Boltzmann-Grad theorem,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 11, 12–16 (1989).

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  65. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev,Mathematical Foundations of Classical Statistical Mechanics, Gordon and Breach, London (1989).

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  66. D. Ya. Petrina and V. I. Gerasimenko, “Existence of the Boltzmann-Grad limit for an infinite system of hard spheres,”Teor. Mat. Fiz.,83, No. 3, 92–114 (1990).

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  67. D. Ya. Petrina and V. I. Gerasimenko, “Mathematical problems of a statistical system of hard spheres,”Usp. Mat. Nauk,45, No. 3, 135–182 (1990).

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  68. D. Ya. Petrina,Exactly Solvable Models of Quantum Statistical Mechanics, Preprint No. 18, Dipartimento di Matematica Politecnico di Torino, Torino (1992).

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  69. D. Ya. Petrina,Mathematical Foundations of Quantum Statistical Mechanics, Kluwer, Dordrecht (1995).

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  70. D. Ya. Petrina,Mathematical Foundations of Quantum Statistical Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995).

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  71. D. Ya. Petrina, V. I. Gerasimenko, and V. Z. Enol'skii, “Equations of motion for a class of quantum-classical systems,”Dokl. Akad Nauk SSSR,315, No. 1, 75–80 (1990).

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    V. I. Gerasimenko, P. V. Malyshev & A. L. Rebenko

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  1. V. I. Gerasimenko
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  2. P. V. Malyshev
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  3. A. L. Rebenko
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 317–328, April, 1994.

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Gerasimenko, V.I., Malyshev, P.V. & Rebenko, A.L. On D. Ya. Petrina's works in contemporary mathematical physics. Ukr Math J 46, 331–342 (1994). https://doi.org/10.1007/BF01060403

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  • DOI: https://doi.org/10.1007/BF01060403

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