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 TMF, 2001, Volume 126, Number 1, Pages 149–163 (Mi tmf421)

Some Properties of Functional Integrals with Respect to the Bogoliubov Measure

D. P. Sankovich

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider problems related to integration with respect to the Bogoliubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. An inequality for traces is proved, and an upper estimate is derived for the Gibbs equilibrium mean square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.

DOI: https://doi.org/10.4213/tmf421

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English version:
Theoretical and Mathematical Physics, 2001, 126:1, 121–135

Bibliographic databases:

Citation: D. P. Sankovich, “Some Properties of Functional Integrals with Respect to the Bogoliubov Measure”, TMF, 126:1 (2001), 149–163; Theoret. and Math. Phys., 126:1 (2001), 121–135

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. Corgini, D. P. Sankovich, “Local Gaussian Dominance: An Anharmonic Excitation of Free Bosons”, Theoret. and Math. Phys., 132:1 (2002), 1019–1028
2. D. P. Sankovich, “The Bogolyubov Functional Integral”, Proc. Steklov Inst. Math., 251 (2005), 213–245
3. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625
4. R. S. Pusev, “Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm”, Theoret. and Math. Phys., 165:1 (2010), 1348–1357
5. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149
6. Nazarov A.I., Sheipak I.A., “Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes”, Bull London Math Soc, 44:1 (2012), 12–24
7. V. R. Fatalov, “Asymptotic behavior of small deviations for Bogoliubov's Gaussian measure in the $L^p$ norm, $2\le p\le\infty$”, Theoret. and Math. Phys., 173:3 (2012), 1720–1733
8. V. R. Fatalov, “Exact Laplace-type asymptotic formulas for the Bogoliubov Gaussian measure: The set of minimum points of the action functional”, Theoret. and Math. Phys., 191:3 (2017), 870–885
9. Lifshits M. Nazarov A., “L-2-Small Deviations For Weighted Stationary Processes”, Mathematika, 64:2 (2018), 387–405
10. Nazarov A.I. Nikitin Ya.Yu., “On Small Deviation Asymptotics in l-2 of Some Mixed Gaussian Processes”, 6, no. 4, 2018, 55
11. 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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