Representation of the Coulomb Matrix Elements by Means of Appell Hypergeometric Function F2

Article

Abstract

Exact analytical representation for the Coulomb matrix elements by means of Appell’s double series F2 is derived. The finite sum obtained for the Appell function F2 allows us to evaluate explicitly the matrix elements of the two-body Coulomb interaction in the lowest Landau level. An application requiring the matrix elements of Coulomb potential in quantum Hall effect regime is presented.

Keywords

Quantum Hall fluid Electron gas Hypergeometric functions Appell functions 

Mathematics Subject Classification (2010)

33Cxx 33C65 

Notes

Acknowledgements

I would like to thank the anonymous reviewers for the valuable suggestions given for improving presentation of the paper. Particularly, I would like to thank greatly the anonymous reviewer who evoked questions that allowed to make clear the deep insights of this paper and its impact on future works within this line of research.

References

  1. 1.
    Ezawa, Z.F.: Quantum Hall Effects, 2nd edn. World Scientific, Singapore (2008)CrossRefMATHGoogle Scholar
  2. 2.
    Ciftja, O.: J. Math. Phys. 52, 122105 (2011)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bailey, W.N.: Generalized hypergeometric series, 2nd edition, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 32. Cambridge University Press, Cambridge (1964)Google Scholar
  4. 4.
    Goerge, G., Mizan, R.: Basic hypergeometric series, 2nd edition, Encyclopedia of Mathematics and its applications, vol. 96. Cambridge University Press, Cambridge (2004)Google Scholar
  5. 5.
    Tarasov, O.V.: I. J. Mod. Phys. 9, 2699 (1995)ADSCrossRefGoogle Scholar
  6. 6.
    Tarasov, O.V.: Nucl. Phys. B 854, 841 (2012)ADSCrossRefGoogle Scholar
  7. 7.
    Appell, P.: C. R. Acad. Séances l’Acad. Sci. 90, 296–298 and 731–735 (1880)Google Scholar
  8. 8.
    Shpot, M.A.: J. Math. Phys. 48, 123512 (2007)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Jain, J.K.: Composite Fermions. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  10. 10.
    Laughlin, R.B.: Phys. Rev. Lett. 50, 1395 (1983)ADSCrossRefGoogle Scholar
  11. 11.
    Jain, J.K.: :Phys. Rev. Lett. 63, 199–202 (1989)Google Scholar
  12. 12.
    Jain, J.K.: Phys. Rev. B 41, 7653 (1990)ADSCrossRefGoogle Scholar
  13. 13.
    Jain, J.K.: Science 266, 1199 (1994)ADSCrossRefGoogle Scholar
  14. 14.
    Goldman, V.J., Tsiper, E.V.: Phys. Rev. Lett. 86, 5841 (2001)ADSCrossRefGoogle Scholar
  15. 15.
    Tsiper, E.V., Goldman, V.J.: Phys. Rev. B 64, 165311 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    Tsiper, E.V.: J. Math. Phys. 43, 1664 (2002)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1980)MATHGoogle Scholar
  18. 18.
    Saad, N., Hall, R.L.: J. Phys. A 36, 7771 (2003)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Slater, L.J.: Generalized Hypergeometric Functions. Combridge University Press, Cambridge (1966)MATHGoogle Scholar
  20. 20.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted/Wiley, New York (1984)MATHGoogle Scholar
  21. 21.
    Ciftja, O.: Physica B 404, 227 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Shakirov, Sh.: Phys. Lett. A 375, 984 (2011)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Theoretical Physics LaboratoryUniversity of TlemcenTlemcenAlgeria

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