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 TMF, 2005, Volume 142, Number 3, Pages 530–555 (Mi tmf1796)

Algebra with polynomial commutation relations for the Zeeman–Stark effect in the hydrogen atom

M. V. Karasev, E. M. Novikova

Moscow State Institute of Electronics and Mathematics

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.

Keywords: integrable systems, nonlinear commutation relations, coherent states, resonance asymptotic behavior of spectrum

DOI: https://doi.org/10.4213/tmf1796

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English version:
Theoretical and Mathematical Physics, 2005, 142:3, 447–469

Bibliographic databases:

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

Citation in format AMSBIB
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• https://doi.org/10.4213/tmf1796
• http://mi.mathnet.ru/eng/tmf/v142/i3/p530

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This publication is cited in the following articles:
1. Efstathiou, K, “Most Typical 12 Resonant Perturbation of the Hydrogen Atom by Weak Electric and Magnetic Fields”, Physical Review Letters, 101:25 (2008), 253003
2. 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
3. M. V. Karasev, E. M. Novikova, “Algebra and quantum geometry of multifrequency resonance”, Izv. Math., 74:6 (2010), 1155–1204
4. Efstathiou K., Sadovskii D.A., “Normalization and global analysis of perturbations of the hydrogen atom”, Rev Modern Phys, 82:3 (2010), 2099–2154
5. 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
6. 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
7. 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
8. A. S. Migaeva, A. V. Pereskokov, “Asimptotika spektra atoma vodoroda v ortogonalnykh elektricheskom i magnitnom polyakh vblizi nizhnikh granits spektralnykh klasterov”, Matem. zametki, 107:5 (2020), 734–751
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