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Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian. (English. Russian original) Zbl 1280.81046
Theor. Math. Phys. 172, No. 1, 987-1000 (2012); translation from Teor. Mat. Fiz. 172, No. 1, 122-137 (2012).
Summary: We determine the rate with which finitely multiple approximations in the Feynman formula converge to the exact expression for the equilibrium density operator of a harmonic oscillator in the linear \(\tau\)-quantization. We obtain an explicit analytic expression for a finitely multiple approximation of the equilibrium density operator and the related Wigner function. We show that in the class of \(\tau\)-quantizations, the equilibrium Wigner function of a harmonic oscillator is positive definite only in the case of the Weyl quantization.

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47D08 Schrödinger and Feynman-Kac semigroups
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[1] P. R. Chernoff, J. Funct. Anal., 2, 238–242 (1968). · Zbl 0157.21501
[2] R. P. Feynman, Rev. Modern Phys., 20, 367–387 (1948). · Zbl 1371.81126
[3] R. P. Feynman, Phys. Rev., 84, 108–128 (1951). · Zbl 0044.23304
[4] O. G. Smolyanov, A. G. Tokarev, and A. Truman, J. Math. Phys., 43, 5161–5171 (2002). · Zbl 1060.58009
[5] E. Nelson, J. Math. Phys., 5, 332–343 (1964). · Zbl 0133.22905
[6] N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. 1, Fourier Analysis and Semigroups, Imperial College Press, London (2001). · Zbl 0987.60003
[7] V. P. Maslov, Complex Markov Chains and the Feynman Path Integral for Nonlinear Equations [in Russian], Nauka, Moscow (1976). · Zbl 0449.35086
[8] P. Cartier and C. De Witt-Morett, Functional Integration: Action and Symmetries, Cambridge University Press, Cambridge (2006). · Zbl 1122.81004
[9] O. G. Smolyanov and E. T. Shavgulidze, Path Integrals [in Russian], Moscow State Univ., Moscow (1990).
[10] S. A. Albeverio, R. J. Høegh-Krohn, and S. Mazzucchi, Mathematical Theory of Feynman Path Integrals (Lect. Notes Math., Vol. 523), Springer, Berlin (2008). · Zbl 1222.46001
[11] M. Kac, Probability and Related Topics in Physical Sciences, Amer. Math. Soc., Providence, R. I. (1959). · Zbl 0087.33003
[12] A. D. Venttsel’ and M. I. Freidlin, Fluctuations in Dynamical Systems Subject to Small Random Perturbations [in Russian], Nauka, Moscow (1979).
[13] E. B. Dynkin, Markov Processes [in Russian], Fizmatlit, Moscow (1963).
[14] Yu. L. Daletskij and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983); English transl. (Math. and Its Appl. Soviet Series, Vol. 76), Kluwer, Dordrecht (1991). · Zbl 0536.46031
[15] F. A. Berezin, Theor. Math. Phys., 6, 141–155 (1971).
[16] H. von Waizsaecker and O. G. Smolyanov, Dokl. Math., 79, 335–338 (2009). · Zbl 1175.81106
[17] M. Gadella and O. G. Smolyanov, Dokl. Math., 77, 120–123 (2008). · Zbl 1159.35426
[18] Ya. A. Butko, O. G. Smolyanov, and R. L. Schilling, Dokl. Math., 82, 679–683 (2010). · Zbl 1213.47047
[19] V. G. Sakbaev and O. G. Smolyanov, Dokl. Math., 82, 630–633 (2010). · Zbl 1200.81054
[20] D. S. Tolstyga, Russian J. Math. Phys., 18, 122–131 (2011). · Zbl 1251.35105
[21] Yu. N. Orlov, Foundations of Quantization of Degenerate Dynamical Systems [in Russian], Moscow Inst. Phys. Tech., Moscow (2004).
[22] Ya. A. Butko, O. G. Smolyanov, Contemp. Probl. Math. Mech., 6, 61–75 (2011).
[23] Ya. A. Butko, M. Grothaus, and O. G. Smolyanov, Dokl. Math., 78, 590–595 (2008). · Zbl 1218.81037
[24] F. A. Berezin, The Method of Second Quantization [in Russian], Nauka, Moscow (1986). · Zbl 0678.58044
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