Geodesic
Polynomial Conservation Laws in Quantum Systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 140 (2004) no. 3, pp. 460-479 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e. the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.
Keywords: Hamiltonian operator, polynomial differential operator, system with exponential interaction, potential spectrum. 
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V. V. Kozlov; D. V. Treschev. Polynomial Conservation Laws in Quantum Systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 140 (2004) no. 3, pp. 460-479. http://geodesic.mathdoc.fr/item/TMF_2004_140_3_a7/

[1] V. V. Kozlov, Teplovoe ravnovesie po Gibbsu i Puankare, Institut kompyuternykh issledovanii, Moskva–Izhevsk, 2002 | MR

[2] V. V. Kozlov, D. V. Treschev, Matem. sb., 135:1 (1988), 119–138

[3] V. V. Kozlov, Dokl. RAN, 373:5 (2000), 597–599 | MR | Zbl

[4] M. Adler, P. van Moerbeke, Adv. Math., 38 (1980), 267–317 ; O. I. Bogoyavlensky, Commun. Math. Phys., 51:3 (1976), 201–209 | DOI | MR | Zbl | DOI | MR

[5] M. Adler, P. van Moerbeke, Commun. Math. Phys., 83 (1982), 83–106 ; M. A. Olshanetsky, A. M. Perelomov, Invent. Math., 37:2 (1976), 93–108 ; 54:3 (1979), 261–269 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[6] V. V. Kozlov, D. V. Treschev, Izv. AN SSSR. Ser. matem., 51:3 (1989), 537–556

[7] S. Gravel, TMF, 137:1 (2003), 97–107 | DOI | MR | Zbl

[8] M. A. Olshanetskii, A. M. Perelomov, A. G. Reiman, M. A. Semenov-Tyan-Shanskii, “Integriruemye sistemy, II”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 16, VINITI, M., 1987, 86–226 | MR

[9] B. Kostant, “Quantization and representation theory”, Representation Theory of Lie Groups, London Math. Soc. Lect. Notes Ser., 34, eds. M. F. Atiyah, G. L. Luke, Cambridge Univ. Press, New York, 1979, 287–316 ; М. А. Семенов-Тян-Шанский, Квантовые цепочки Тоды. Теоремы разложения и рассеяние, Препринт ЛОМИ, 1984, с. 3–84; R. Goodman, N. R. Wallach, Commun. Math. Phys., 83 (1982), 355–386 ; 94 (1984), 177–217 | MR | DOI | MR | Zbl | DOI | MR | Zbl

[10] P. K. Rashevskii, Uchen. zapiski Mosk. ped. in-ta im. K. Libktekhta. Ser. fiz.-matem. nauk, 1938, no. 2, 83–94