Geodesic
On a family of commutative Wick symbols
Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 1, pp. 38-49 
A. A. Tsvetkov. On a family of commutative Wick symbols. Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 1, pp. 38-49. http://geodesic.mathdoc.fr/item/TMF_1981_47_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Conditions are found under which an infinite system of bosons with two-body interaction has a one-parameter family of integrals of the motion determined by the Wick symbol $$ A(\lambda)=\exp\biggl[\dfrac1\lambda\mathbf P+\dfrac{h}{2\lambda^2}\mathbf V \biggr], $$ where $\mathbf V$ and $\mathbf P$ are the Wick symbols of the interaction and momentum operatore, respectively. Examples of interaction potentials for which these conditions are satisfied are given. The complete integrability of the corresponding classical systems is proved.

[1] Bethe H., “Zur Theorie der Metall. 1: Eigenwerte und Eigenfunktionen, Atomkette”, Zeitschrift für Physik, 71 (1931), 205–226 | DOI | Zbl

[2] Berezin F. A., Pokhil G. P., Finkelberg V. M., “Uravnenie Shredingera dlya sistemy odnomernykh chastits s tochechnym vzaimodeistviem”, Vestn. MGU, matematika, mekhanika, 1964, no. 1, 21–26 | MR

[3] Tsvetkov A. A., “Ob integralakh dvizheniya bozonov s tochechnym vzaimodeistviem”, Vestn. MGU, matematika, mekhanika, 1977, no. 4, 61–69 | Zbl

[4] Zakharov V. E., Shabat A. B., “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61 (1971), 118–134 | MR

[5] Mozer J., “Three integrable Hamiltonian systems connected with isospectral deformations”, Adv. Math., 16:2 (1975), 197–220 | DOI | MR

[6] Calogero F., “Exactly solvable one-dimensional many-body problems”, Lett Nuovo Cim., 13:11 (1975), 411–416 | DOI | MR

[7] Pidkuiko S. I., Stepin A. M., “Polinomialnye integraly gamiltonovykh sistem”, Dokl. AN SSSR, 239:1 (1978), 50–53 | MR | Zbl

[8] Perelomov A. M., “Tochnye rezultaty dlya odnomernykh mnogochastichnykh sistem”, Fizika elementarnykh chastits i atomnogo yadra, 10:4 (1979), 850–883 | MR

[9] Manakov S. V., O tochnom reshenii sistemy nelineinykh uravnenii Shredingera, Preprint 26-73, IYaF, Novosibirsk, 1973

[10] Berezin F. A., Metod vtorichnogo kvantovaniya, Nauka, M., 1965 | MR

[11] Gelfand I. M., Dikii A. A., “Rezolventy i gamiltonovy sistemy”, Funkts. analiz i ego prilozh., 11 (1977), 11–27 | Zbl

[12] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii, t. 3, Nauka, M., 1967 | MR