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 TMF, 1995, Volume 104, Number 2, Pages 310–329 (Mi tmf1340)

Complex germ method in the Fock space. I. Asymptotics like wave packets

V. P. Maslov, O. Yu. Shvedov

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: In this paper, we establish a new method of constructing approximate solutions to secondary-quantized equations, for instance, for many-particle Schrödinger and Liouville equations written in terms of the creation and annihilation operators, and also for equations of quantum field theory. The method is based on transformation of these equations to an infinite-dimensional Schrödinger equation, which is investigated by semiclassical methods. We use, and generalize to the infinite-dimensional case, the complex germ method, which yields wave packet type asymptotics in the Schrödinger representation. We find the corresponding asymptotics in the Fock space and show that the state vectors obtained are actually asymptotic solutions to secondary-quantized equations with an accuracy $O(\varepsilon ^{M/2})$, $M\in \mathbb N$, with respect to the parameter $\varepsilon$ of the semiclassical expansion.

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English version:
Theoretical and Mathematical Physics, 1995, 104:2, 1013–1028

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Citation: V. P. Maslov, O. Yu. Shvedov, “Complex germ method in the Fock space. I. Asymptotics like wave packets”, TMF, 104:2 (1995), 310–329; Theoret. and Math. Phys., 104:2 (1995), 1013–1028

Citation in format AMSBIB
\Bibitem{MasShv95} \by V.~P.~Maslov, O.~Yu.~Shvedov \paper Complex germ method in the Fock space. I. Asymptotics like wave packets \jour TMF \yr 1995 \vol 104 \issue 2 \pages 310--329 \mathnet{http://mi.mathnet.ru/tmf1340} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1488677} \zmath{https://zbmath.org/?q=an:0855.35107} \transl \jour Theoret. and Math. Phys. \yr 1995 \vol 104 \issue 2 \pages 1013--1028 \crossref{https://doi.org/10.1007/BF02065981} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995UD33400009} 

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This publication is cited in the following articles:
1. V. P. Maslov, O. Yu. Shvedov, “Complex germ method in the Fock space. II. Asymptotics, corresponding to finite-dimensional isotropic manifolds”, Theoret. and Math. Phys., 104:3 (1995), 1141–1161
2. V. P. Maslov, “Sufficient Conditions for High-Temperature Superconductivity”, Funct. Anal. Appl., 29:4 (1995), 286–288
3. V. P. Maslov, O. Yu. Shvedov, “Initial conditions in quasi-classical field theory”, Theoret. and Math. Phys., 114:2 (1998), 184–197
4. V. P. Maslov, O. Yu. Shvedov, “Asymptotics of the density matrix of a system of a large number of identical particles”, Math. Notes, 65:1 (1999), 70–88
5. Maslov V.P., Shvedov O.Y., “Large-N expansion as a semiclassical approximation to the third-quantized theory”, Physical Review D, 60:10 (1999), 105012
6. V. P. Maslov, A. E. Ruuge, “Many-particle and semiclassical limit transitions for nonrelativistic bosons in a quantized electromagnetic field”, Theoret. and Math. Phys., 125:3 (2000), 1687–1701
7. V. P. Maslov, O. Yu. Shvedov, “The Complex-Germ Method for Statistical Mechanics of Model Systems”, Proc. Steklov Inst. Math., 228 (2000), 234–251
8. Shvedov, OY, “Time evolution in an external field: The unitarity paradox”, Annals of Physics, 287:2 (2001), 260
9. Alexey Borisov, Alexander Shapovalov, Andrey Trifonov, “Transverse Evolution Operator for the Gross–Pitaevskii Equation in Semiclassical Approximation”, SIGMA, 1 (2005), 019, 17 pp.
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