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Russian Physics Journal

, Volume 61, Issue 9, pp 1645–1652 | Cite as

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

  • Yu. A. GrishechkinEmail author
  • V. N. Kapshai
Article
  • 5 Downloads

Approximate analytical and numerical solutions of the three-dimensional Logunov–Tavkhelidze equation are found for the spherically symmetric case. Solutions are obtained in momentum and relativistic configuration representations. The wave functions in the relativistic configuration representation have additional zeroes compared to the wave functions of the nonrelativistic harmonic oscillator in the coordinate representation.

keywords

quasipotential equation harmonic oscillator relativistic configuration representation momentum representation Sturm–Liouville problem integral equation MacDonald function hypergeometric series energy spectrum wave function 

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References

  1. 1.
    A. M. L. Messiah, Quantum mechanics, Vol. 1 [Russian translation], Nauka, Moscow (1978).Google Scholar
  2. 2.
    A. I. Baz’, Ya. B. Zeldovich, and A. M. Perelomov, Scattering, Reactions, and Decays in Nonrelativistic Quantum Mechanics [in Russian], Nauka, Moscow (1971).Google Scholar
  3. 3.
    N. M. Atakishiyev, R. M. Mir-Kasimov, and Sh. M. Nagiyev, Ann. Phys., 42, No. 1, 25–30 (1985).CrossRefGoogle Scholar
  4. 4.
    S. M. Nagiyev, E. I. Jafarov, and R. M. Imanov, J. Phys. A, 36, 7813–7824 (2003).ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    S. M. Nagiyev, E. I. Jafarov, R. M. Imanov, and L. Homorodean, Phys. Lett. A, 334, 260–266 (2005).ADSCrossRefGoogle Scholar
  6. 6.
    R. A. Frick, Ann. Phys., 523, No. 11, 871–882 (2011).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. A. Logunov and A. N. Tavkhelidze, Nuovo Cimento, 29, No. 2, 380–399 (1963).CrossRefGoogle Scholar
  8. 8.
    V. G. Kadyshevskii, R. M. Mir-Kasimov, and N. B. Skachkov, Elem. Chast. Atomn. Yadra, 2, No. 3, 635–690 (1972).Google Scholar
  9. 9.
    V. N. Kapshai and S. I. Fialka, Russ. Phys. J., 60, No. 10, 1696–1704 (2017).CrossRefGoogle Scholar
  10. 10.
    V. N. Kapshai and T. A. Alferova, J. Phys. A, 32, 5329–5342 (1999).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Fizmatlit, Moscow (2008).Google Scholar
  12. 12.
    M. Abramowits and I. Stigan, Handbook of Special Functions with Formulas, Graphs and Mathematical Tables [in Russian], Nauka, Moscow (1979).Google Scholar
  13. 13.
    N. W. McLachlan, Theory and Applications of Mathieu Functions [Russian translation], Inostrannaya Literatura, Moscow (1953).Google Scholar
  14. 14.
    N. N. Kalitkin, Numerical Methods [in Russian], BHV-Petersburg, Saint Petersburg (2011).Google Scholar
  15. 15.
    H. Bateman and A. Erdélui, Higher Transcendental Functions, Vol. 1 [Russian translation], Nauka, Moscow (1973).Google Scholar
  16. 16.
    Yu. A. Grishechkin and V. N. Kapshai, Russ. Phys. J., 56, No. 4, 435–443 (2013).CrossRefGoogle Scholar
  17. 17.
    V. N. Kapshai, K. P. Shilyaeva, and Yu. A. Grishechkin, Probl. Fiz., Matem. Tekh., 9, No. 4, 33–37 (2011).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Francisk Skorina Gomel State UniversityGomelBelarus

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