Geodesic
Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom
Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 3, pp. 530-555 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the Zeeman–Stark effect for the hydrogen atom in crossed homogeneous electric and magnetic fields. A nonhomogeneous perturbing potential can also be present. If the crossed fields satisfy some resonance relation, then the degeneration in the resonance spectral cluster is removed only in the second-order term of the perturbation theory. The averaged Hamiltonian in this cluster is expressed in terms of generators of some dynamical algebra with polynomial commutation relations; the structure of these relations is determined by a pair of coprime integers contained in the resonance ratio. We construct the irreducible hypergeometric representations of this algebra. The averaged spectral problem in the irreducible representation is reduced to a second-or third-order ordinary differential equation whose solutions are model polynomials. The asymptotic behavior of the solution of the original problem concerning the Zeeman–Stark effect in the resonance cluster is constructed using the coherent states of the dynamical algebra. We also describe the asymptotic behavior of the spectrum in nonresonance clusters, where the degeneration is already removed in the first-order term of the perturbation theory.
Keywords: integrable systems, nonlinear commutation relations, coherent states, resonance asymptotic behavior of spectrum. 
@article{TMF_2005_142_3_a4,
     author = {M. V. Karasev and E. M. Novikova},
     title = {Algebra with polynomial commutation relations for the {Zeeman{\textendash}Stark} effect in the hydrogen atom},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {530--555},
     year = {2005},
     volume = {142},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a4/}
}
TY  - JOUR
AU  - M. V. Karasev
AU  - E. M. Novikova
TI  - Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2005
SP  - 530
EP  - 555
VL  - 142
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a4/
LA  - ru
ID  - TMF_2005_142_3_a4
ER  - 
%0 Journal Article
%A M. V. Karasev
%A E. M. Novikova
%T Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2005
%P 530-555
%V 142
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a4/
%G ru
%F TMF_2005_142_3_a4
M. V. Karasev; E. M. Novikova. Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom. Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 3, pp. 530-555. http://geodesic.mathdoc.fr/item/TMF_2005_142_3_a4/

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

[2] M. V. Karasev, E. M. Novikova, TMF, 141:3 (2004), 424 | DOI | MR | Zbl

[3] M. V. Karasev, E. M. Novikova, TMF, 142:1 (2005), 127 | DOI | MR | Zbl

[4] M. J. Gourlay, T. Uzer, D. Farrelly, Phys. Rev. A, 47 (1993), 3113 ; J. von Milczewski, G. H. F. Diercksen, T. Uzer, Phys. Rev. Lett., 73 (1994), 2428 ; T. Uzer, D. Farrelly, Phys. Rev. A, 52 (1995), R2501 | DOI | DOI | DOI | MR

[5] D. A. Sadovskií, B. I. Ẑhilinskií, L. Michel, Phys. Rev. A, 53 (1996), 4064 | DOI | MR

[6] C. Neumann, R. Ubert, S. Freund et al., Phys. Rev. Lett., 78 (1997), 4705 | DOI

[7] M. V. Karasev, E. M. Novikova, TMF, 108:3 (1996), 339 | DOI | MR | Zbl

[8] P. S. Laplace, Traité de mecanique celeste, Bachelier, Paris, 1829

[9] M. V. Karasev, E. M. Novikova, “Non-Lie permutation relations, coherent states, and quantum embedding”, Coherent Transform, Quantization, and Poisson Geometry, ed. M. V. Karasev, AMS, Providence, RI, 1998, 1–202 | MR

[10] M. V. Karasev, E. M. Novikova, Matem. zametki, 70:6 (2001), 854 | DOI | MR | Zbl

[11] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, T. 1, Nauka, M., 1973 | MR

[12] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR

[13] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, T. 2, Nauka, M., 1974 | MR