Abstract
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.
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REFERENCES
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods, Mir, Moscow (1976).
M. V. Karasev and E. M. Novikova, Theor. Math. Phys., 141, 1698 (2004).
M. V. Karasev and E. M. Novikova, Theor. Math. Phys., 142, 109 (2005).
M. J. Gourlay, T. Uzer, and D. Farrelly, Phys. Rev. A, 47, 3113 (1993); J. von Milczewski, G. H. F. Diercksen, and T. Uzer, Phys. Rev. Lett., 73, 2428 (1994); T. Uzer and D. Farrelly, Phys. Rev. A, 52, R2501 (1995).
D. A. Sadovskii, B. I. Zhilinskii, and L. Michel, Phys. Rev. A, 53, 4064 (1996).
C. Neumann et al., Phys. Rev. Lett., 78, 4705 (1997).
M. V. Karasev and E. M. Novikova, Theor. Math. Phys., 108, 1119 (1996).
P. S. Laplace, Traité de mecanique celeste, Bachelier, Paris (1829).
M. V. Karasev and E. M. Novikova, “Non-Lie permutation relations, coherent states, and quantum embedding, ” in: Coherent Transform, Quantization, and Poisson Geometry (Amer. Math. Soc. Transl. Ser. 2, Vol. 187, M. V. Karasev, ed.), Amer. Math. Soc., Providence, R. I. (1998), p. 1.
M. V. Karasev and E. M. Novikova, Math.Notes, 70, 779 (2001).
H. Bateman and A. Erdélyi, eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 1, McGraw-Hill, New York (1953).
E. Kamke, Differentialgleichungen, Lösunsmethoden, und Lösungen, Acad. Verlag, Leipzig (1959).
H. Bateman and A. Erdélyi, eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 2, McGraw-Hill, New York (1953).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 530–555, March, 2005.
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Karasev, M.V., Novikova, E.M. Algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom. Theor Math Phys 142, 447–469 (2005). https://doi.org/10.1007/s11232-005-0035-8
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DOI: https://doi.org/10.1007/s11232-005-0035-8