Geodesic
Quantum tops as examples of commuting differential operators
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 3, pp. 355-374 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the quantum analogues of tops on the Lie algebras $so(4)$ and $e(3)$ represented by differential operators.
Mots-clés : quantization
Keywords: integrable top, operator algebra. 
@article{TMF_2012_172_3_a2,
     author = {V. E. Adler and V. G. Marikhin and A. B. Shabat},
     title = {Quantum tops as examples of commuting differential operators},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {355--374},
     year = {2012},
     volume = {172},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_3_a2/}
}
TY  - JOUR
AU  - V. E. Adler
AU  - V. G. Marikhin
AU  - A. B. Shabat
TI  - Quantum tops as examples of commuting differential operators
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2012
SP  - 355
EP  - 374
VL  - 172
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2012_172_3_a2/
LA  - ru
ID  - TMF_2012_172_3_a2
ER  - 
%0 Journal Article
%A V. E. Adler
%A V. G. Marikhin
%A A. B. Shabat
%T Quantum tops as examples of commuting differential operators
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2012
%P 355-374
%V 172
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2012_172_3_a2/
%G ru
%F TMF_2012_172_3_a2
V. E. Adler; V. G. Marikhin; A. B. Shabat. Quantum tops as examples of commuting differential operators. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 3, pp. 355-374. http://geodesic.mathdoc.fr/item/TMF_2012_172_3_a2/

[1] I. Schur, Sitzungsber. Berl. Math. Ges., 4 (1905), 2–8 | Zbl

[2] J. L. Burchnall, T. W. Chaundy, Proc. R. Soc. Lond. A, 118:780 (1928), 557–583 | DOI | Zbl

[3] S. P. Novikov, Funkts. analiz i ego pril., 8:3 (1974), 54–66 | DOI | MR | Zbl

[4] I. M. Krichever, Funkts. analiz i ego pril., 12:3 (1978), 20–31 | DOI | MR | Zbl

[5] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, UMN, 42:4(256) (1987), 3–53 | DOI | MR | Zbl

[6] R. A. Gabiev, A. B. Shabat, TMF, 171:1 (2012), 18–25 | DOI | DOI | MR | Zbl

[7] A. P. Veselov, Dokl. AN SSSR, 270:6 (1983), 1298–1300 | MR | Zbl

[8] A. G. Reyman, M. A. Semenov-Tian-Shansky, Commun. Math. Phys., 105:3 (1986), 461–472 | DOI | MR | Zbl

[9] V. V. Sokolov, Dokl. RAN, 394:5 (2004), 602–605 | MR | Zbl

[10] V. G. Marikhin, V. V. Sokolov, Regul. Chaotic Dyn., 10:1 (2005), 59–70 | DOI | MR | Zbl

[11] F. Schottky, Berl. Ber., 1891, 227–232 | Zbl

[12] S. V. Manakov, Funkts. analiz i ego pril., 10:4 (1976), 93–94 | DOI | MR | Zbl

[13] W. Stekloff, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3), 1 (1909), 145–256 | MR | Zbl

[14] M. Adler, P. van Moerbeke, Adv. Math. Suppl. Stud., 9 (1986), 81–96 | MR | Zbl

[15] V. V. Sokolov, “Generalized Kowalewski top: new integrable cases on $e(3)$ and $so(4)$,”, The Kowalevski Property, CRM Proceedings and Lecture Notes, 32, ed. V. B. Kuznetsov, AMS, Providence, RI, 2002, 307–313, arXiv: nlin/0110022 | DOI | MR | Zbl

[16] I. V. Komarov, V. B. Kuznetsov, J. Phys. A, 24:13 (1991), L737–L742 | DOI | MR | Zbl

[17] A. Clebsch, Math. Ann., 3:2 (1870), 238–262 | DOI | MR | Zbl

[18] S. V. Kowalevski, Acta Math., 12:1 (1889), 177–232 | DOI | MR | Zbl

[19] O. Laporte, Phys. Rev., 43:7 (1933), 548–551 | DOI | Zbl

[20] I. V. Komarov, TMF, 47:1 (1981), 67–72 | DOI | MR

[21] I. V. Komarov, TMF, 50:3 (1982), 402–409 | DOI | MR

[22] I. V. Komarov, V. B. Kuznetsov, TMF, 73:3 (1987), 335–347 | DOI | MR

[23] I. V. Komarov, J. Phys. A, 34:11 (2001), 2111–2120 | DOI | MR | Zbl

[24] A. Ramani, B. Grammaticos, B. Dorizzi, Phys. Lett. A, 101:2 (1984), 69–71 | DOI | MR

[25] A. V. Borisov, I. S. Mamaev, Dinamika tverdogo tela. Gamiltonovy metody, integriruemost, khaos, IKI, M., Izhevsk, 2005 | MR

[26] J. Hietarinta, Phys. Lett. A, 246:1–2 (1998), 97–104 | DOI | MR | Zbl

[27] M. A. Olshanetsky, A. M. Perelomov, Phys. Rept., 94:6 (1983), 313–404 | DOI | MR

[28] L. D. Landau, E. M. Lifshits, Kurs teoreticheskoi fiziki, v. 3, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1989 | MR | Zbl

[29] V. G. Marikhin, V. V. Sokolov, Regul. Chaotic Dyn., 15:6 (2010), 652–658 | DOI | MR | Zbl

[30] H. A. Kramers, G. P. Ittmann, Z. Physik, 53:7–8 (1929), 553–565 | DOI | Zbl

[31] M-P. Grosset, A. P. Veselov, Proc. Edinb. Math. Soc. Ser. 2, 51:3 (2008), 635–650 | DOI | MR | Zbl