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.
Citation:
M. V. Karasev, E. M. Novikova, “Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom”, TMF, 142:3 (2005), 530–555; Theoret. and Math. Phys., 142:3 (2005), 447–469
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\by M.~V.~Karasev, E.~M.~Novikova
\paper Algebra with polynomial commutation relations for the Zeeman--Stark effect in the hydrogen atom
\jour TMF
\yr 2005
\vol 142
\issue 3
\pages 530--555
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\jour Theoret. and Math. Phys.
\yr 2005
\vol 142
\issue 3
\pages 447--469
\crossref{https://doi.org/10.1007/s11232-005-0035-8}
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Linking options:
https://www.mathnet.ru/eng/tmf1796
https://doi.org/10.4213/tmf1796
https://www.mathnet.ru/eng/tmf/v142/i3/p530
This publication is cited in the following 14 articles:
A. V. Pereskokov, “Asymptotics of hypergeometric coherent states and eigenfunctions of the hydrogen atom in a magnetic field. Determination of self-consistent energy levels”, Theoret. and Math. Phys., 222:3 (2025), 453–470
A. V. Pereskokov, “Semiclassical Asymptotics of the Spectrum of the Hydrogen Atom in an Electromagnetic Field Near the Lower Boundaries of Spectral Clusters”, J Math Sci, 259:2 (2021), 244
A. S. Migaeva, A. V. Pereskokov, “Asymptotics of the Spectrum of the Hydrogen Atom in Orthogonal Electric and Magnetic Fields near the Lower Boundaries of Spectral Clusters”, Math. Notes, 107:5 (2020), 804–819
A. S. Migaeva, A. V. Pereskokov, “Semiclassical Asymptotics of the Spectrum of the Hydrogen Atom in an Electromagnetic Field Near the Upper Boundaries of Spectral Clusters”, J Math Sci, 251:6 (2020), 850
Pereskokov A.V., “On the Asymptotics of the Spectrum of the Hydrogen Atom in Orthogonal Electric and Magnetic Fields Near the Upper Boundaries of Spectral Clusters”, Russ. J. Math. Phys., 26:3 (2019), 391–400
E. M. Novikova, “Algebra of Symmetries of Three-Frequency Hyperbolic Resonance”, Math. Notes, 106:6 (2019), 940–956
Francesco Fassò, Daniele Fontanari, Dmitrií A. Sadovskií, “An Application of Nekhoroshev Theory to the Study of the Perturbed Hydrogen Atom”, Math Phys Anal Geom, 18:1 (2015)
A. V. Pereskokov, “Asymptotics of the spectrum and quantum averages of a perturbed resonant oscillator near the boundaries of spectral clusters”, Izv. Math., 77:1 (2013), 163–210
A. V. Pereskokov, “Asymptotics of the Spectrum and Quantum Averages near the Boundaries of Spectral Clusters for Perturbed Two-Dimensional Oscillators”, Math. Notes, 92:4 (2012), 532–543
A. V. Pereskokov, “Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters”, Trans. Moscow Math. Soc., 73 (2012), 221–262
M. V. Karasev, E. M. Novikova, “Algebra and quantum geometry of multifrequency resonance”, Izv. Math., 74:6 (2010), 1155–1204
Efstathiou K., Sadovskii D.A., “Normalization and global analysis of perturbations of the hydrogen atom”, Rev Modern Phys, 82:3 (2010), 2099–2154
Efstathiou, K, “Complete classification of qualitatively different perturbations of the hydrogen atom in weak near-orthogonal electric and magnetic fields”, Journal of Physics A-Mathematical and Theoretical, 42:5 (2009), 055209
Efstathiou, K, “Most Typical 12 Resonant Perturbation of the Hydrogen Atom by Weak Electric and Magnetic Fields”, Physical Review Letters, 101:25 (2008), 253003
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