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Creative Contribution of D. Ya. Petrina to the Development of Contemporary Mathematical Physics

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Abstract

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

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REFERENCES

  1. O. S. Parasyuk, D. Ya. Petrina, and P. N. Tatsunyak, “The Källén – 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 in the problem of diffraction,” Ukr. Mat. Zh.,10, No. 3, 405–412 (1958).

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

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

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  5. D. Ya. Petrina and V. I. Kolomytsev, “On one addition to the Bogolyubov–Vladimirov theorem,” Ukr. Mat. Zh.,12, No. 2, 165–169 (1960).

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  6. D. Ya. Petrina, “On the impossibility of the construction 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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  7. 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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  8. 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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  9. D. Ya. Petrina, “Analytic properties of the contributions of Feynman diagrams,” Dokl. Akad. Nauk SSSR,149, No. 4, 808–811(1963).

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  10. 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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  11. 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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  12. D. Ya. Petrina, “Mandelstam representation and the continuity theorem,” Zh. éksp. Teor. Fiz.,46, Issue 2, 544–554 (1964).

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  13. D. Ya. Petrina, “A proof of the Mandelstam representation for a ladder diagram of the sixth order,” Zh. éksp. Teor. Fiz.,47,Issue 2, 524–529 (1964).

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

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

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

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  18. 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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  22. D. Ya. Petrina, “On the 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 the S-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. élement. Chastits 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,” Dokl. Akad. Nauk Ukr. SSR, Ser. A,No. 8,756–760 (1976).

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

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

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

  35. 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 con-nection 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, P. V. Malyshev, and A. I. Pilyavskii, “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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  42. D. Ya. Petrina and A. I. Pilyavskii, “Potential of the electrostatic field of a system of charged particles and a dynamical mem-brane,” Dokl. Akad. Nauk Ukr. SSR, Ser. A,No. 7, 57–60 (1981).

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

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

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

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  51. D. Ya. Petrina, “Distribution functions of systems of charged particles in spatially-inhomogeneous media,” Teor. Mat. Fiz.,59, No. 1, 104–116 (1984).

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  52. D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic limit of nonequilibrium states of a three-dimensional system of hard spheres,” in: Proceedings of the 2nd International Conference on Selected Problems of Statistical Mechanics[in Russian], Vol. 2, Dubna (1984), pp. 152–159.

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

  54. D. Ya. Petrina, “Mathematical description of the evolution of infinite systems of classical statistical physics. Locally perturbed one-dimensional systems,” in: Proceedings of the 2nd International Workshop[in Russian], Part 1, Naukova Dumka, Kiev (1985), pp. 109–117.

  55. D. Ya. Petrina and A. I. Pilyavskii, “Problems of electrostatics in spatially-inhomogeneous media and the subtraction procedure,” Fiz. Mnogochastich. Sist.,Issue 7, 82–96 (1985).

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  56. V. I. Gerasimenko and D. Ya. Petrina, “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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  57. 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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  59. Yu. A. Mitropol'skii, D. Ya. Petrina, and V. G. Bar'yakhtar, “Creative contribution of Academician N. N. Bogolyubov to the de-velopment of mathematics, nonlinear mechanics, and theoretical physics,” Vestn. Akad. Nauk Ukr. SSSR,No. 11, 9–21 (1985).

  60. 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. 228–237.

  61. D. Ya. Petrina and V. I. Gerasimenko, “Evolution of states of infinite systems in classical mechanics,” in: Sov. Sci. Rev., Sec. C,Math. Phys. Rev.,Vol. 5, Harwood, New York (1985), pp. 1–52.

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  62. Yu. A. Mitropol'skii, V. G. Bar'yakhtar, and D. Ya. Petrina, “Science is the main and unique aim of my life,” Nauk. Kul't.,20, 128–137 (1986).

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  63. V. I. Gerasimenko, P. V. Malyshev, and D. Ya. Petrina, “Solutions of the Bogolyubov equations for infinite three-dimensional systems of particles,” in: Proceedings of the 4th International Vilnius Conference on Probability Theory and Mathematical Sta-tistics, Vol. 2, Vilnius (1987), pp. 451–461.

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  64. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, “Thermodynamic limit for solutions of the Bogolyubov equations,” Sov. Sci. Rev., Ser. C, Mat. Phys.,7, 281–337 (1988).

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  65. D. Ya. Petrina, “Mathematical problems of the description of the evolution of states of infinite systems of statistical mechanics,” in: “Selected Problems of Statistical Physics,”Vol. 2, World Scientific, Singapore (1987), pp. 332–342.

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  66. 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, 336–341 (1987).

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

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

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

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

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

  72. D. Ya. Petrina, “Mathematical problems of the description of the evolution of states of infinite systems of statistical mechanics,” in: Plasma Theory and Nonlinear Turbulent Processes in Physics,Vol. 2, (1988), pp. 906–932

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  73. D. Ya. Petrina, P. V. Malyshev, and V. I. Gerasimenko, “Mathematical problems of the description of the evolution of states of infinite classical statistical systems,” in: Proceedings of the 3rd International Workshop on Nonlinear and Turbulent Processes in Physics[in Russian], Vol. 1, Naukova Dumka, Kiev (1988), pp. 312–315.

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  74. D. Ya. Petrina, “On the process of flow of an electrolyte through a membrane as a boundary-value problem in domains with fine-grained structure,” in: Collection of Scientific Works[in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1988), pp. 4–14.

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

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  77. D. Ya. Petrina and V. I. Gerasimenko, “Boltzmann – Grad limit for solutions of the Bogolyubov equations,” in: Proceedings of the 4th International Workshop on Nonlinear and Turbulent Processes in Physics[in Russian], Vol. 2, Naukova Dumka, Kiev(1989), pp. 57–60.

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  78. V. S. Vladimirov, Yu. A. Mitropol'skii, O. S. Parasyuk, A. M. Samoilenko, and D. Ya. Petrina, “Investigations of N. N. Bogo-lyubov in mathematics and theoretical physics,” Ukr. Mat. Zh.,41, No. 9, 1156–1165 (1989).

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  79. D. Ya. Petrina, “Thermodynamic limit of solutions of the Bogolyubov equations,” Tr. Mat. Inst. Akad. Nauk SSSR,191, 192–201 (1989).

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  81. V. I. Gerasimenko and D. Ya. Petrina, “Existence of the Boltzmann–Grad limit for an infinite system of hard spheres,” Teor. Mat. Fiz.,83, No. 1, 92–114 (1990).

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

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  84. D. Ya. Petrina, “On Hamiltonians in spaces of translation-invariant functions,” in: Methods of Mathematical Physics of Infinite Systems[in Russian], Kiev (1991), pp. 4–21.

  85. Yu. A. Mitropol'skii and D. Ya. Petrina, “On N. N. Bogolyubov's works in classical and quantum statistical mechanics,” Ukr. Mat. Zh.,45, No. 2, 155–201 (1993).

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  86. D. Ya. Petrina, “Approximation of a general Hamiltonian by Hamiltonians of the theory of superconductivity and superfluidity,” Algebraic and Geometric Methods in Mathematical Physics,Netherlands (1994).

  87. D. Ya. Petrina, “General Hamiltonians and model Hamiltonians of the theory of superconductivity and superfluidity in the Hilbert spaces of translation-invariant functions,” in: Operator Theory: Advances and Applications,Vol. 70, Birkhäuser (1994), pp. 213–217.

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

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

  90. D. Ya. Petrina and V. I. Gredzhuk, “On two models on interacting Bose gas: Bogolyubov's model of superfluidity and Huang– Yang–Luttinger model,” Phys. Math. Rev.,Nos. 3–4, 370–406 (1996).

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  91. S. Cercignani, V. Gerasimenko, and D. Ya. Petrina, Many Particle Dynamics and Kinetic Equations,Kluwer, Dordrecht (1997).

  92. D. Ya. Petrina and E. D. Petrina, “Existence of equilibrium states of systems of hard spheres in the Boltzmann – Enskog limit within the framework of grand canonical ensemble,” Ukr. Mat. Zh.,49, No. 1, 112–121 (1997).

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  93. M. Lampis and D. Ya. Petrina, “Boltzmann – Enskog limit for equilibrium states of systems of hard spheres within the frameworkof canonical ensemble,” Ukr. Mat. Zh.,49, No. 9, 1194–1205 (1997).

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  94. V. I. Gerasimenko and D. Ya. Petrina, “On the generalized kinetic equation,” Dokl. Akad. Nauk Ukr. SSR,No. 7, 7–12 (1997).

  95. D. Ya. Petrina, “Thermodynamic limit for nonequilibrium states of classical statistical systems,” in: Mathematical Physics. En-cyclopedia[in Russian], Bol'shaya Sovetskaya éntsiklopediya, Moscow (1998), pp. 599–600.

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  96. D. Ya. Petrina and K. D. Petrina, “Stochastic dynamics and Boltzmann hierarchy. I,” Ukr. Mat. Zh.,50, No. 2, 195–210 (1998).

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  97. D. Ya. Petrina and K. D. Petrina, “Stochastic dynamics and Boltzmann hierarchy. II,” Ukr. Mat. Zh.,50, No. 3, 372–387 (1998).

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  98. D. Ya. Petrina and K. D. Petrina, “Stochastic dynamics and Boltzmann hierarchy. III,” Ukr. Mat. Zh.,50, No. 4, 552–569 (1998).

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  99. V. I. Gerasimenko and D. Ya. Petrina, “The generalized kinetic equation generated by the BBGKY hierarchy,” Ukr. Fiz. Zh.,43, Nos. 6–7, 697–702 (1998).

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  100. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems,Taylor and Francis, London (2002).

  101. M. Lampis, D. Ya. Petrina, and K. D. Petrina, “Stochastic dynamics as a limit of Hamiltonian dynamics of hard spheres,” Ukr. Mat. Zh.,51, No. 5, 614–635 (1999).

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  102. D. Ya. Petrina, “Methods for derivation of the stochastic Boltzmann hierarchy,” Ukr. Mat. Zh.,52, No. 4, 474–491 (2000).

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  103. D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. I. Spectrum,” Ukr. Mat. Zh.,52, No. 5, 667–690 (2000).

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  104. D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. II. Spectra of excitations,” Ukr. Mat. Zh.,53, No. 8, 1080–1101 (2001).

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  105. M. Lampis and D. Ya. Petrina, “Spatially-homogeneous Boltzmann hierarchy as averaged spatially-inhomogeneous stochastic Boltzmann hierarchy,” Ukr. Mat. Zh.,54, No. 1, 78–94 (2002).

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  106. D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. III. The BCS Hamiltonian with mean-field in-teraction,” Ukr. Mat. Zh.,54, No. 11, 1486–1504 (2002).

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  107. D. Ya. Petrina, “Model BCS Hamiltonian and approximating Hamiltonian in the case of infinite volume. IV. Two branches of their common spectra and states,” Ukr. Mat. Zh.,55, No. 2, 174–196 (2003).

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  108. D. Ya. Petrina, “States of infinite equilibrium classical systems,” Ukr. Mat. Zh.,55, No. 3, 389–400 (2003).

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  109. D. Ya. Petrina, “Equilibrium and nonequilibrium states of the model Fröhlich – Peierls Hamiltonian,” Ukr. Mat. Zh.,55, No. 8, 1069–1087 (2003).

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

    V. I. Herasymenko & P. V. Malyshev

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Herasymenko, V.I., Malyshev, P.V. Creative Contribution of D. Ya. Petrina to the Development of Contemporary Mathematical Physics. Ukrainian Mathematical Journal 56, 357–373 (2004). https://doi.org/10.1023/B:UKMA.0000045684.38326.dc

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