Skip to main content
SpringerLink
Log in

How to relate the oscillator and Coulomb systems on spheres and pseudospheres?

  • Symposium on Integrable Systems
  • Published:
Physics of Atomic Nuclei Aims and scope Submit manuscript

Abstract

We show that oscillators on a sphere and a pseudosphere are related, by the so-called Bohlin transformation, with Coulomb systems on a pseudosphere: even states of an oscillatoryield a conventional Coulomb system on a pseudosphere, while odd states yield a Coulomb system on a pseudosphere in the presence of a magnetic flux tube generating half-spin. In higher dimensions, oscillator and Coulomb(-like) systems are connected in similar way. In particular, applying the Kustaanheimo-Stiefel transformation to oscillators on a sphere and a pseudosphere, we obtained a pseudospherical generalization of the MIC-Kepler problem describing a three-dimensional charge-dyon system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. M. Ter-Antonyan, quant-ph/0003106.

  2. K. Bohlin, Bull. Astron. 28, 144 (1911); T. Levi-Civita, Opere Math. 2, 411 (1906).

    Google Scholar 

  3. P. Kustaanheimo and E. Stiefel, J. Reine Angew. Math. 218, 204 (1965).

    MathSciNet  Google Scholar 

  4. A. Hurwitz, Mathematische Werke (Birkhäuser, Basel, 1933), Vol. II, p. 641; L. S. Davtyan et al., J. Phys. A 20, 6121 (1987); D. Lambert and M. Kibler, J. Phys. A 21, 307 (1988).

    Google Scholar 

  5. A. Nersessian, V. M. Ter-Antonyan, and M. Tsulaia, Mod. Phys. Lett. A 11, 1605 (1996); A. Nersessian and V. M. Ter-Antonyan, Yad. Fiz. 61, 1868 (1998) [Phys. At. Nucl. 61, 1756 (1998)].

    ADS  MathSciNet  Google Scholar 

  6. T. Iwai and Y. Uwano, J. Math. Phys. (N.Y.) 27, 1523 (1986); J. Phys. A 21, 4083 (1988); A. Nersessian and V. Ter-Antonyan, Mod. Phys. Lett. A 9, 2431 (1994); 10, 2633 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  7. T. Iwai, J. Geom. Phys. 7, 507 (1990); L. G. Mardoyan, A. N. Sissakian, and V. M. Ter-Antonyan, hep-th/9601093; Yad. Fiz. 61, 1859 (1998) [Phys. At. Nucl. 61, 1746 (1998)]; hep-th/9803010; T. Iwai and T. Sunako, J. Geom. Phys. 20, 250 (1996); 30, 326 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  8. E. Schrödinger, Proc. R. Ir. Soc. 46, 9, 183 (1941); 47, 53 (1941).

    Google Scholar 

  9. P. W. Higgs, J. Phys. A 12, 309 (1979); H. I. Leemon, J. Phys. A 12, 489 (1979).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. A. Barut, A. Inomata, and G. Junker, J. Phys. A 20, 6271 (1987); 23, 1179 (1990); Ya. A. Granovsky, A. S. Zhedanov, and I. M. Lutzenko, Teor. Mat. Fiz. 91, 207, 396 (1992); D. Bonatsos, C. Daskaloyanis, and K. Kokkotas, Phys. Rev. A 50, 3700 (1994); C. Grosche, G. S. Pogosyan, and A. N. Sissakian, Fortschr. Phys. 43, 523 (1995); E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Math. Phys. (N.Y.) 37, 6439 (1996); 38, 5416 (1997).

    ADS  MathSciNet  Google Scholar 

  11. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Math. Phys. (N.Y.) 41, 2629 (2000); Preprint IMA 1680 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Nersessian and G. Pogosyan, Phys. Rev. A 63, 20103(R) (2001); quant-ph/0006118.

  13. D. Zwanziger, Phys. Rev. 176, 1480 (1968); H. McIntosh and A. Cisneros, J. Math. Phys. (N.Y.) 11, 896 (1970).

    ADS  Google Scholar 

  14. V. V. Gritsev, Yu. A. Kurochkin, and V. S. Otchik, J. Phys. A 33, 4903 (2000).

    ADS  MathSciNet  Google Scholar 

  15. C. N. Yang, J. Math. Phys. (N.Y.) 19, 320 (1978).

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980, Russia

    A. Nersessian

Authors
  1. A. Nersessian
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

From Yadernaya Fizika, Vol. 65, No. 6, 2002, pp. 1103–1108.

Original English Text Copyright © 2002 by Nersessian.

This article was submitted by the author in English.

1) On leave of absence from Yerevan State University, ul. A. Manougian 1, 375025 Yerevan, Armenia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nersessian, A. How to relate the oscillator and Coulomb systems on spheres and pseudospheres?. Phys. Atom. Nuclei 65, 1070–1075 (2002). https://doi.org/10.1134/1.1490113

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/1.1490113

Keywords

Search

Navigation