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

This article is cited in 8 scientific papers (total in 8 papers)

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:

Received: 12.04.2004

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  adsnasa  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. M. V. Karasev, E. M. Novikova, “Algebra and quantum geometry of multifrequency resonance”, Izv. Math., 74:6 (2010), 1155–1204  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Efstathiou K., Sadovskii D.A., “Normalization and global analysis of perturbations of the hydrogen atom”, Rev Modern Phys, 82:3 (2010), 2099–2154  crossref  adsnasa  isi  elib  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet  zmath  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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