Geodesic
Algebra with polynomial commutation relations for the Zeeman effect in the Coulomb–Dirac field
Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 1, pp. 127-147 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a model of particle motion in the field of an electromagnetic monopole (in the Coulomb–Dirac field) perturbed by homogeneous and inhomogeneous electric fields. After quantum averaging, we obtain an integrable system whose Hamiltonian can be expressed in terms of the generators of an algebra with polynomial commutation relations. We construct the irreducible representations of this algebra and its hypergeometric coherent states. We use these states to represent the eigenfunctions of the original problem in terms of the solutions of the model ordinary differential equation. We also present the asymptotic approximations of the eigenvalues in the leading term of the perturbation theory, where the degeneration of the spectrum is removed completely.
Keywords: integrable systems, nonlinear commutation relations, coherent states, asymptotic behavior of spectrum.
Mots-clés : Dirac monopole 
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M. V. Karasev; E. M. Novikova. Algebra with polynomial commutation relations for the Zeeman effect in the Coulomb–Dirac field. Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 1, pp. 127-147. http://geodesic.mathdoc.fr/item/TMF_2005_142_1_a11/

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

[2] M. V. Karasev, E. M. Novikova, TMF, 141:3 (2004), 424–454 | DOI | MR | Zbl

[3] M. V. Karasev, E. M. Novikova, UMN, 49:5 (1994), 169–170 | MR | Zbl

[4] M. V. Karasev, E. M. Novikova, TMF, 108:3 (1996), 339–387 | DOI | MR | Zbl

[5] H. V. McIntosh, A. Cisneros, J. Math. Phys., 11:3 (1970), 896–916 ; I. Mladenov, V. Tsanov, J. Phys. A, 20 (1987), 5865–5871 ; T. Iwai, Y. Uwano, J. Phys. A, 21 (1988), 4083–4104 ; I. Mladenov, Ann. Inst. H. Poincare, 50 (1989), 219–227 ; A. Yoshioka, K. Ii, J. Math. Phys., 31:6 (1990), 1388–1394 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[6] M. V. Karasev, E. M. Novikova, “Adapted connections, Hamilton dynamics, geometric phases and quantization over isotropic submanifolds”, Coherent Transform, Quantization, and Poisson Geometry, ed. M. V. Karasev, AMS Publ., Providence, RI, 1998, 1–202 | MR

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

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

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