Physics of Atomic Nuclei

, Volume 76, Issue 3, pp 365–375 | Cite as

Parabolic sturmians approach to the three-body continuum Coulomb problem

Nuclei Theory


The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e , e , He++) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Bray and A. T. Stelbovics, Phys. Rev. Lett. 70, 746 (1993).ADSCrossRefGoogle Scholar
  2. 2.
    I. Bray, D. V. Fursa, A. S. Kheifets, and A. T. Stelbovics, J. Phys. B 35, R117 (2002).ADSCrossRefGoogle Scholar
  3. 3.
    Z. Papp, Phys. Rev. C 55, 1080 (1997).ADSCrossRefGoogle Scholar
  4. 4.
    Z. Papp, C.-Y. Hu, Z.T. Hlousek, et al., Phys. Rev. A 63, 062721 (2001).ADSCrossRefGoogle Scholar
  5. 5.
    Yu. V. Popov, S. A. Zaytsev, and S. I. Vinitsky, Phys. Part. Nucl. 42, 683 (2011).CrossRefGoogle Scholar
  6. 6.
    The J-Matrix Method: Developments and Applications, Ed. by A. D. Alhaidari, E. J. Heller, H.A. Yamani, and M. S. Abdelmonem (Springer Sci., BusinessMedia, 2008).Google Scholar
  7. 7.
    S. A. Zaytsev, V. A. Knyr, Yu. V. Popov, and A. Lahmam-Bennani, Phys. Rev. A 75, 022718 (2007).ADSCrossRefGoogle Scholar
  8. 8.
    M. Silenou Mengoue, M. G. Kwato Njock, B. Piraux, Yu. V. Popov, and S. A. Zaytsev, Phys. Rev. A 83, 052708 (2011).ADSCrossRefGoogle Scholar
  9. 9.
    K. Bartschat, E. T. Hudson, M. P. Scott, et al., J. Phys. B 29, 115 (1996).ADSCrossRefGoogle Scholar
  10. 10.
    T. N. Rescigno, M. Baertschy, W. A. Isaacs, and C. W. McCurdy, Science 286, 2474 (1999).CrossRefGoogle Scholar
  11. 11.
    P. L. Bartlett, A. T. Stelbovics, and I. Bray, J. Phys. B 37, L69 (2004).ADSCrossRefGoogle Scholar
  12. 12.
    M. S. Pindzola and D. R. Schultz, Phys. Rev. A 53, 1525 (1996).ADSCrossRefGoogle Scholar
  13. 13.
    V. V. Serov, V. L. Derbov, B. B. Joulakian, and S. I. Vinitsky, Phys. Rev. A 78, 063403 (2008).ADSCrossRefGoogle Scholar
  14. 14.
    V. V. Serov, V. L. Derbov, B. B. Joulakian, and S. I. Vinitsky, Phys. Rev. A 75, 012715 (2007).ADSCrossRefGoogle Scholar
  15. 15.
    A. L. Frapiccini, J. M. Randazzo, G. Gasaneo, and F. D. Colavecchia, J. Phys. B 43, 101001 (2010).ADSCrossRefGoogle Scholar
  16. 16.
    A. S. Kadyrov, I. Bray, A. M. Mukhamedzhanov, and A. T. Stelbovics, Phys. Rev. Lett. 101, 230405 (2008).ADSCrossRefGoogle Scholar
  17. 17.
    S. A. Zaytsev, J. Phys. A 41, 265204 (2008).MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    S. A. Zaytsev, J. Phys. A 42, 015202 (2009).MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    S. A. Zaytsev, J. Phys. A 43, 385208 (2010).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    H. Klar, Z. Phys. D 16, 231 (1990).ADSCrossRefGoogle Scholar
  21. 21.
    L. Rosenberg, Phys. Rev. D 8, 1833 (1973).ADSCrossRefGoogle Scholar
  22. 22.
    Dz. Belkic, J. Phys. B 11, 3529 (1978).ADSCrossRefGoogle Scholar
  23. 23.
    C. R. Garibotti and J. E. Miraglia, Phys. Rev. A 21, 572 (1980).ADSCrossRefGoogle Scholar
  24. 24.
    M. Brauner, J. S. Briggs, and H. Klar, J. Phys. B 22, 2265 (1989).ADSCrossRefGoogle Scholar
  25. 25.
    S. Jones and D. H. Madison, Phys. Rev. Lett. 91, 073201 (2003).ADSCrossRefGoogle Scholar
  26. 26.
    L. U. Ancarani, T. Montagnese, and C. Dal Capello, Phys. Rev. A 70, 012711 (2004).ADSCrossRefGoogle Scholar
  27. 27.
    O. Chuluunbaatar, H. Bachau, Yu. V. Popov, et al., Phys. Rev. A 81, 063424 (2010).ADSCrossRefGoogle Scholar
  28. 28.
    P. A. Macri, J. E. Miraglia, C. R. Garibotti, et al., Phys. Rev. A 55, 3518 (1997).ADSCrossRefGoogle Scholar
  29. 29.
    P. C. Ojha, J.Math. Phys. 28, 392 (1987).MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    C. Canuto, A. Quarteroni, M. Y. Hussaini, and T. A. Zang, Spectral Methods. Fundamentals in Single Domains (Springer-Verlag, Berlin, Heidelberg, 2006).MATHGoogle Scholar
  31. 31.
    M. Stone, Mathematics for Physics I (Pimander-Casaubon, Alexandria, Florence, London, 2002).Google Scholar
  32. 32.
    J. Révai, M. Sotona, and J. Žofka, J. Phys. G 11, 745 (1985).ADSCrossRefGoogle Scholar
  33. 33.
    B. Kónya, G. Lévai, and Z. Papp, Phys. Rev. C 61, 034302 (2000).ADSCrossRefGoogle Scholar
  34. 34.
    L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particle Systems (Kluwer Academic, Dordrecht, 1993).MATHCrossRefGoogle Scholar
  35. 35.
    S. P. Merkuriev, J. Math. Sci. 22, 1638 (1983).CrossRefGoogle Scholar
  36. 36.
    J. Berakdar, Phys. Rev. A 53, 2314 (1996).ADSCrossRefGoogle Scholar
  37. 37.
    R. A. Swainson and G. W. Drake, J. Phys. A 24, 95 (1991).MathSciNetADSMATHCrossRefGoogle Scholar
  38. 38.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).Google Scholar
  39. 39.
    V. S. Buslaev and S. B. Levin, arXiv:1104.3358v1 [math-ph].Google Scholar
  40. 40.
    M. V. Fedoryuk, The Saddle Point Method (Nauka, Moscow, 1977) [in Russian].MATHGoogle Scholar
  41. 41.
    V. Hutson and J. S. Pym, Applications of Functional Analysis and Operator Theory (Academic Press, London, New York, Toronto, Sydney, San Francisco, 1980).MATHGoogle Scholar
  42. 42.
    Z. Papp, Phys. Rev. A 46, 4437 (1992).ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Pacific National UniversityKhabarovskRussia
  2. 2.Nuclear Physics InstituteMoscow State UniversityMoscowRussia
  3. 3.Institute of Condensed Matter and NanosciencesUniversité catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations