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Hamiltonian Feynman path integrals via the Chernoff formula. (English) Zbl 1060.58009
Summary: The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (= Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.

MSC:
58D30 Applications of manifolds of mappings to the sciences
46G12 Measures and integration on abstract linear spaces
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
81S40 Path integrals in quantum mechanics
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References:
[1] Feynman, Rev. Mod. Phys. 20 pp 367– (1948)
[2] Feynman, Phys. Rev. 84 pp 108– (1951)
[3] O. G. Smolyanov and E. T. Shavgulidze,Functional Integrals(Moscow State University Press, Moscow, 1990, in Russian).
[4] O. G. Smolyanov and E. T. Shavgulidze, Proceedings of the Fourth Vilnius Conference. Resume of talks, Vol. 3, 1985.
[5] O. G. Smolyanov and E. T. Shavgulidze, Proceedings of the Fourth Vilnius Conference Vol. 2, 1987, p. 595. · Zbl 0654.60047
[6] Smolyanov, Theor. Math. Phys. 100 pp 803– (1994)
[7] Ktitarev, Mat. Zametki 42 pp 40– (1987)
[8] Smolyanov, Math. Notes 68 pp 668– (2000)
[9] Berezin, Teor. Mat. Fiz. 6 pp 194– (1971)
[10] Daubechies, J. Math. Phys. 28 pp 85– (1987)
[11] Chernoff, J. Funct. Anal. 84 pp 238– (1968)
[12] Nelson, J. Math. Phys. 5 pp 332– (1964)
[13] S. Albeverio and R. Hoegh-Krohn,Mathematical Theory of Feynman Path Integrals, Lecture Notes in Math Vol. 523 (Springer, Berlin, 1976). · Zbl 0337.28009
[14] V. P. Maslov,Complex Markov Chains and the Feynman Path Integral for Nonlinear Equations(Nauka, Moscow, 1976). · Zbl 0449.35086
[15] Albeverio, J. Math. Phys. 36 pp 2135– (1995)
[16] Elworthy, Ann. I.H.P. Phys. Theor. 41 pp 115– (1984)
[17] Tokarev, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 2 pp 16– (2001)
[18] G. W. Johnson and M. L. Lapidus,The Feynman Integral and Feynman’s Operational Calculus(Clarendon, Oxford, 2000). · Zbl 0952.46044
[19] Alimov, Teor. Mat. Fiz. 11 pp 182– (1972)
[20] Evgrafov, Dokl. Akad. Nauk SSSR 191 pp 979– (1970)
[21] Maslov, Mod. Probl. Math. 8 pp 137– (1977)
[22] Smolyanov, Theor. Math. Phys. 119 pp 677– (1999)
[23] Smolyanov, Dokl. Math. 64 pp 147– (2001)
[24] Zastawniak, Bull. Pol. Acad. Sci., Chem. 36 pp 341– (1988)
[25] A. G. Tokarev, Analytical and Numerical Methods in Mathematics and Mechanics, Proceedings of the XXII Conference of Young Scientists of the Faculty of Mechanics and Mathematics in the Moscow State University in 17–22 April, 2000. (Moscow State University Press, Moscow, 2001), pp. 156–158.
[26] N. Dunford and J. T. Schwartz,Linear Operators. Part I. General Theory(Wiley Interscience, New York, 1958).
[27] H. Schaefer,Topological Vector Spaces(Springer, New York, 1971). · Zbl 0212.14001
[28] M. Reed and B. Simon,Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness(Academic, New York, 1975). · Zbl 0308.47002
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