Geodesic
Method of calculation of the spectrum of a centrally symmetric Hamiltonian on the basis of approximate $O_4$ and $SU_3$ symmetry
Teoretičeskaâ i matematičeskaâ fizika, Tome 8 (1971) no. 1, pp. 23-36 
V. B. Serebrennikov; A. E. Shabad. Method of calculation of the spectrum of a centrally symmetric Hamiltonian on the basis of approximate $O_4$ and $SU_3$ symmetry. Teoretičeskaâ i matematičeskaâ fizika, Tome 8 (1971) no. 1, pp. 23-36. http://geodesic.mathdoc.fr/item/TMF_1971_8_1_a2/
@article{TMF_1971_8_1_a2,
     author = {V. B. Serebrennikov and A. E. Shabad},
     title = {Method of calculation of the spectrum of a~centrally symmetric {Hamiltonian} on the basis of approximate $O_4$ and $SU_3$ symmetry},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {23--36},
     year = {1971},
     volume = {8},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1971_8_1_a2/}
}
TY  - JOUR
AU  - V. B. Serebrennikov
AU  - A. E. Shabad
TI  - Method of calculation of the spectrum of a centrally symmetric Hamiltonian on the basis of approximate $O_4$ and $SU_3$ symmetry
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1971
SP  - 23
EP  - 36
VL  - 8
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1971_8_1_a2/
LA  - ru
ID  - TMF_1971_8_1_a2
ER  - 
%0 Journal Article
%A V. B. Serebrennikov
%A A. E. Shabad
%T Method of calculation of the spectrum of a centrally symmetric Hamiltonian on the basis of approximate $O_4$ and $SU_3$ symmetry
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1971
%P 23-36
%V 8
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1971_8_1_a2/
%G ru
%F TMF_1971_8_1_a2

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

For classical centrally symmetric problems with an arbitrary potential a study is made of all the integrals of motion that are situated in the plane of the orbit and, together with the angular momentum, form the closed Lie algebra of the groups $O_4$ and $SU_3$ (in the sense of the Poisson brackets). A solution is found to the problem of the unique construction of the invariant Casimir operators from these integrals of motion. A study is made of the problem of quantization and a method (quasiclassical in nature) is proposed for calculating the discrete spectrum of a Hamiltonian by means of the Casimir operators without recourse to the Schrödinger equation. The multiplet structure of the energy levels of the Schrödinger problem is described.

[1] L. Hulten, Z. Phys., 86 (1933), 21 ; V. Fock, Z. Phys., 98 (1935), 145 ; V. Bargmann, Z. Phys., 99 (1936), 376 | DOI | DOI | Zbl | DOI

[2] F. L. Hill, I. M. Iuch, Phys. Rev., 57 (1940), 641 ; Ю. Н. Демков, Вестник ЛГУ, 11 (1953); C. A. Backer, Phys. Rev., 103 (1956), 1119 ; D. Fradkin, Amer. J. Phys., 33 (1965), 207 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[3] A. A. Makarov, J. A. Smorodinsky, Kh. Valiev, P. Winternitz, Nuovo Cim., 52 (1967), 1061 | DOI

[4] H. Bacry, Nuovo Cim., 41 (1966), 222 | DOI | Zbl

[5] C. Fronsdal, Infinite multiplets and the hydrogen atom, preprint, Univ. of California, 1966

[6] D. Fradkin, Progr. Theor. Phys., 37 (1967), 798 | DOI | Zbl

[7] L. C. Biedenharn, Phys. Lett., 3 (1962), 69 | DOI | MR | Zbl

[8] A. M. Perelomov, V. S. Popov, Pisma v ZhETF, 1, 6 ; (1965), 15; ЯФ, 3 (1966), 924 | Zbl | MR