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 Trudy Mat. Inst. Steklova, 2005, Volume 251, Pages 223–256 (Mi tm52)

The Bogolyubov Functional Integral

D. P. Sankovich

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Problems of integration with respect to a special Gaussian measure (the Bogolyubov measure) that arises in the statistical equilibrium theory for quantum systems are considered. It is shown that the Gibbs equilibrium means of Bose operators can be represented as functional integrals with respect to this measure. Certain functional integrals with respect to the Bogolyubov measure are calculated. Approximate formulas are constructed that are exact for functional polynomials of a given degree, as well as formulas that are exact for integrable functionals of a wider class. The nondifferentiability of Bogolyubov trajectories in the corresponding function space is established. A theorem on the quadratic variation of trajectories is proved. The properties of scale transformations that follow from this theorem are studied. Examples of semigroups associated with the Bogolyubov measure are constructed. Independent increments for this measure are found. A relation between the Bogolyubov measure and parabolic partial differential equations is considered. An inequality for traces is proved, and an upper estimate is obtained for the Gibbs equilibrium mean of the square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 251, 213–245

Bibliographic databases:
UDC: 517.958+517.987

Citation: D. P. Sankovich, “The Bogolyubov Functional Integral”, Nonlinear dynamics, Collected papers, Trudy Mat. Inst. Steklova, 251, Nauka, MAIK «Nauka/Inteperiodika», M., 2005, 223–256; Proc. Steklov Inst. Math., 251 (2005), 213–245

Citation in format AMSBIB
\Bibitem{San05} \by D.~P.~Sankovich \paper The Bogolyubov Functional Integral \inbook Nonlinear dynamics \bookinfo Collected papers \serial Trudy Mat. Inst. Steklova \yr 2005 \vol 251 \pages 223--256 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm52} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2234384} \zmath{https://zbmath.org/?q=an:1118.82006} \transl \jour Proc. Steklov Inst. Math. \yr 2005 \vol 251 \pages 213--245 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Prykarpatsky A., Sankovich D., “Nikolai Nikolayevich Bogolubov (Jr.) Foreword”, Condensed Matter Physics, 13:4 (2010), 40101
2. Prykarpatsky A.K., “Reminiscences of unforgettable times of my collaboration with Nikolai N. Bogolubov (Jr.) Foreword”, Condensed Matter Physics, 13:4 (2010), 40102
3. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149
4. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646
5. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81
6. Sankovich D.P., “Gibbs Equilibrium Averages and Bogolyubov Measure”, Problems of Atomic Science and Technology, 2012, no. 1, 248–252
7. A. I. Nazarov, R. S. Pusev, “Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms”, St. Petersburg Math. J., 25:3 (2014), 455–466
8. V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389
9. V. R. Fatalov, “Functional integrals for the Bogoliubov Gaussian measure: Exact asymptotic forms”, Theoret. and Math. Phys., 195:2 (2018), 641–657
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