Geodesic
Complex germ method in the Fock space. I. Asymptotics like wave packets
Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 2, pp. 310-329 
V. P. Maslov; O. Yu. Shvedov. Complex germ method in the Fock space. I. Asymptotics like wave packets. Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 2, pp. 310-329. http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a8/
@article{TMF_1995_104_2_a8,
     author = {V. P. Maslov and O. Yu. Shvedov},
     title = {Complex germ method in the {Fock} space. {I.} {Asymptotics} like wave packets},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {310--329},
     year = {1995},
     volume = {104},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a8/}
}
TY  - JOUR
AU  - V. P. Maslov
AU  - O. Yu. Shvedov
TI  - Complex germ method in the Fock space. I. Asymptotics like wave packets
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1995
SP  - 310
EP  - 329
VL  - 104
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a8/
LA  - ru
ID  - TMF_1995_104_2_a8
ER  - 
%0 Journal Article
%A V. P. Maslov
%A O. Yu. Shvedov
%T Complex germ method in the Fock space. I. Asymptotics like wave packets
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1995
%P 310-329
%V 104
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a8/
%G ru
%F TMF_1995_104_2_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Maslov V. P., Shvedov O. Yu., TMF, 98:2 (1994), 266–288 | MR | Zbl

[2] Maslov V. P., Shvedov O. Yu., Russian J. Math. Phys., 2:2 (1994), 217–234 | MR | Zbl

[3] Maslov V. P., Shvedov O. Yu., DAN, 338:1 (1994), 15–18 | MR | Zbl

[4] Berezin F. A., Metod vtorichnogo kvantovaniya, Nauka, M., 1986 | MR | Zbl

[5] Shveber S., Vvedenie v relyativistskuyu kvantovuyu teoriyu polya, IL, M., 1963

[6] Landau L. D., Lifshits E. M., Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1989 | MR

[7] Schönberg M., Nuovo Cim., 9:12 (1952), 1139 | DOI | MR | Zbl

[8] Schönberg M., Nuovo Cim., 10:4 (1953), 419 | DOI | MR | Zbl

[9] Maslov V. P., Tariverdiev S. E., Teoriya veroyatnostei, matematicheskaya statistika, teoreticheskaya kibernetika, 19, VINITI, M., 1982, 85–125 | MR

[10] Maslov V. P., Operatornye metody, Nauka, M., 1973 | MR

[11] Maslov V. P., Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nauka, M., 1977 | MR

[12] Belov V. V., Dobrokhotov S. Yu., TMF, 92:2 (1992), 215–254 | MR

[13] Saimon B., Model $P(\Phi )_2$ evklidovoi kvantovoi teorii polya, Mir, M., 1976

[14] Glimm Dzh., Dzhaffe A., Matematicheskie metody kvantovoi fiziki: podkhod s ispolzovaniem funktsionalnykh integralov, Mir, M., 1984 | MR | Zbl

[15] Giiemin V., Stenberg S., Geometricheskie asimptotiki, Mir, M., 1981 | MR

[16] Karasev M. V., Maslov V. P., Nelineinye skobki Puassona. Geometriya i kvantovanie, Nauka, M., 1991 | MR | Zbl

[17] Maslov V. P., Shvedov O. Yu., Berezin memorial volume, Advances in Soviet Mathematics (to appear)

[18] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965 | MR

[19] Cooper F., Pi S., Stancioff P., Phys. Rev. D, 34 (1986), 3831 | DOI

[20] Cooper F., Mottola E., Phys. Rev. D, 36 (1987), 3114 | DOI

[21] Pi S., Samiullah M., Phys. Rev. D, 36 (1987), 3128 | DOI | MR