Advertisement

The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1779–1794 | Cite as

Comments on employing the Riesz-Feller derivative in the Schrödinger equation

  • B. Al-SaqabiEmail author
  • L. Boyadjiev
  • Yu. Luchko
Regular Article

Abstract

In this paper, we deal with a fractional Schrödinger equation that contains the quantum Riesz-Feller derivative instead of the Laplace operator in the case of a particle moving in a potential field. In particular, this equation is solved for a free particle in terms of the Fox H-function. On the other hand, we show that from physical viewpoint, the fractional Schrödinger equation with the quantum Riesz-Feller derivative of order α, 0 < α ≤ 2 and skewness θ makes sense only if it reduces to the Laplace operator (α = 2) or to the quantum Riesz fractional derivative (θ = 0). The reason is that the quantum Riesz-Feller derivative is a Hermitian operator and possesses real eigenvalues only when α = 2 or θ = 0. We then focus on the time-independent one-dimensional fractional Schrödinger equation with the quantum Riesz derivative in the case of a particle moving in an infinite potential well. In particular, we show that the explicit formulas for the eigenvalues and eigenfunctions of the time-independent fractional Schrödinger equation that some authors recently claimed to receive cannot be valid. The problem to find right formulas is still open.

Keywords

European Physical Journal Special Topic Laplace Operator Fractional Derivative Free Particle Quantum Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Dong, M. Xu, J. Math. Anal. Appl. 344, 1005 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    W. Feller, On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, 73 (1952)Google Scholar
  3. 3.
    R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)Google Scholar
  4. 4.
    R. Gorenflo, F. Mainardi, Fract. Calc. Appl. Anal. 1, 167 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    R. Gorenflo, F. Mainardi, J. Anal. Appl. 18, 231 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    X. Guo, M. Xu, J. Math. Phys. 47, 082104 (2006)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    A.A. Kilbas, Yu.F. Luchko, H. Martinez, J.J. Trujillo, Integral Trans. Special Funct. 21, 779 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Kwasnicki, J. Funct. Anal. 262, 2379 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. Laskin, Fractals Quantum Mech. Chaos 10, 780 (2000)Google Scholar
  10. 10.
    N. Laskin, Fractional Quant. Mech. Lévy Path Integr. Phys. Lett. A 268, 298 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    N. Laskin, Phys. Rev. E 66, 056108 (2002)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    N. Laskin, Principles of fractional quantum mechanics, edited by J. Klafter, S.C. Lim and R. Metzler, Fractional Dynamics: Recent Advances (World Scientific, Singapore, 2012), p. 393Google Scholar
  13. 13.
    Yu. Luchko, H. Matrinez, J.J. Trujillo, Fract. Calc. Appl. Anal. 11, 457 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    F. Mainardi, Yu. Luchko, G. Pagnini, Fract. Calc. Appl. Anal. 4, 153 (2001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-function. Theory and Applications (Berlin, Springer, 2010)Google Scholar
  16. 16.
    I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)Google Scholar
  17. 17.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publ., London, 1993)Google Scholar
  18. 18.
    Sh. Wang, M. Xu, J. Math. Phys. 48, 043502 (2007)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005)Google Scholar
  20. 20.
    A. Zoia, A. Rosso, M. Kardar, Phys. Rev. E 76, 021116 (2007)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKuwait UniversitySafatKuwait
  2. 2.Department of Mathematics, Physics, and ChemistryBeuth Technical University of Applied Sciences BerlinBerlinGermany

Personalised recommendations