Skip to main content
SpringerLink
Log in

Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

The fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space derivative, and solve this equation for an infinite potential by using the Adomian decomposition method. The infinite domain solution of the space-time fractional Schrödinger equation in the case of Riesz space fractional derivative is obtained in terms of the Fox H-functions. We interpret our results for the fractional Schrödinger equation by introducing a complex effective potential in the standard Schrödinger equation, which can be used to describe quantum transport in quantum dots.

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. K. Zhou and Y. Cherruault, Convergence of Adomian’s method applied to nonlinear equations. Math. Comput. Modelling 20 (1994), 69–73.

    MathSciNet  MATH  Google Scholar 

  2. K. Zhou and Y. Cherruault, New ideas for proving convergence of decomposition methods. Comput. Math. Appl. 29 (1995), 103–108.

    MathSciNet  MATH  Google Scholar 

  3. G. Adomian, Stochastic Systems. Academic Press, New York (1983).

    MATH  Google Scholar 

  4. G. Adomian, Nonlinear Stochastic Operator Equations. Academic Press, New York (1986).

    MATH  Google Scholar 

  5. G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Publishers, Dordrecht (1988).

    MATH  Google Scholar 

  6. Q.M. Al-Mdallal, An efficient method for solving fractional Sturm- Liouville problems. Chaos, Solitons and Fractals 40 (2009), 183–189.

    MathSciNet  MATH  Google Scholar 

  7. G.A. Baraff, Model for the effect of finite phase-coherence length on resonant transmission and capture by quantum wells. Phys. Rev. B 58 (1998), # 13799.

    Google Scholar 

  8. S.S. Bayin, Comment on “On the consistency of the solutions of the space fractional Schrödinger equation”. J. Math. Phys. 53 (2012), # 042105.

    MathSciNet  MATH  Google Scholar 

  9. S.S. Bayin, Time fractional Schrödinger equation: Fox’s H-functions and the effective potential. J. Math. Phys. 54 (2013), # 012103.

    MathSciNet  MATH  Google Scholar 

  10. J. Bisquert, Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous-time random walk. Phys. Rev. Lett. 91 (2003), # 010602.

    Google Scholar 

  11. J. Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination. Phys. Rev. E 72 (2005), # 011109.

    Google Scholar 

  12. E. Capelas de Zhou, F.S. Costa, and J. Vaz Jr., The fractional Schrödinger equation for delta potentials. J. Math. Phys. 51 (2010), # 123517.

    MathSciNet  MATH  Google Scholar 

  13. E. Capelas de Oliveira and J. Vaz Jr., Tunneling in fractional quantum mechanics. J. Phys. A: Math. Theor. 44 (2011), # 185303.

    MathSciNet  MATH  Google Scholar 

  14. M. Caputo, Elasticità e Dissipazione. Zanichelli, Bologna (1969).

    Google Scholar 

  15. Y. Zhou and G. Adomian, Decomposition methods: A new proof of convergence. Math. Comput. Modelling 18 (1993), 103–106.

    MathSciNet  MATH  Google Scholar 

  16. J. Dong, Green’s function for the time-dependent scattering problem in the fractional quantum mechanics. J. Math. Phys. 52 (2011), # 042103.

    MathSciNet  MATH  Google Scholar 

  17. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York (1968).

    MATH  Google Scholar 

  18. D.K. Zhou, J.R. Baker, and R. Akis, Complex potentials, dissipative processes, and general quantum transport. In: Technical Proc. of 1999 Internat. Conf. on Modelling and Simulation of Micro Systems, NSTI (1999), 373–376.

    Google Scholar 

  19. H.J. Zhou, A.M. Mathai, and R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011 (2011), # 298628.

    Google Scholar 

  20. R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011).

    MATH  Google Scholar 

  21. R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).

    MATH  Google Scholar 

  22. R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284 (2002), 399–408.

    Google Scholar 

  23. R. Hilfer, On Fractional relaxation. Fractals 11 (2003), 251–257.

    MathSciNet  MATH  Google Scholar 

  24. R. Hilfer, Foundations of fractional dynamics: A short account. In: Fractional Dynamics, Recent Advances, World Scientific, Singapore (2011), 209-227.

    Google Scholar 

  25. R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B 104 (2000), 3914–3917.

    Google Scholar 

  26. A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80 (2009), 022103.

    Google Scholar 

  27. A. Iomin, Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos, Solitons and Fractals 44 (2011), 348–352.

    MathSciNet  MATH  Google Scholar 

  28. A. Iomin, Lévy flights in a box. Chaos, Solitons and Fractals 71 (2015), 73–77.

    MathSciNet  MATH  Google Scholar 

  29. M. Jeng, S.-L.-Y. Xu, E Hawkins, J.M. Schwarz, On the nonlocality of the fractional Schrödinger equation. J. Math. Phys. 51 (2010), # 062102.

    MathSciNet  MATH  Google Scholar 

  30. X. Zhou, H. Qi, and M. Xu, Exact solutions of fractional Schrödingerlike equation with a nonlocal term. J. Math. Phys. 52 (2011), # 042105.

    MathSciNet  MATH  Google Scholar 

  31. L.D. Zhou and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, New York (1977).

    Google Scholar 

  32. N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000), 298–305.

    MathSciNet  MATH  Google Scholar 

  33. N. Laskin, Fractional quantum mechanics. Phys. Rev. E 62 (2000), # 3135.

    MATH  Google Scholar 

  34. N. Laskin, Fractional Schrödinger equation. Phys. Rev. E 66 (2002), # 056108.

    MathSciNet  Google Scholar 

  35. E.K. Zhou, M.K. Zhou, R. Rossato, and L.C.M. Filho, Solutions for diffusion equation with a nonlocal term. Acta Scientiarum. Technology 31 (2009), 81–86.

    Google Scholar 

  36. E.K. Zhou, H.V. Zhou, H. Mukai, and R.S. Mendes, Continuoustime random walk as a guide to fractional Schrödinger equation. J. Math. Phys. 51 (2010), # 092102.

    MathSciNet  Google Scholar 

  37. E.K. Zhou, H.V. Ribeiro, M.A.F. dos Zhou, R. Rossato, and R.S. Mendes, Time dependent solutions for a fractional Schrödinger equation with delta potentials. J. Math. Phys. 54 (2013), # 082107.

    MathSciNet  MATH  Google Scholar 

  38. E.K. Zhou, B.F. de Zhou, L.R. da Silva, and L.R. Evangelista, Solutions for a Schrödinger equation with a nonlocal term. J. Math. Phys. 49 (2008), # 032108.

    MathSciNet  MATH  Google Scholar 

  39. Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys. 54 (2013), # 012111.

    MathSciNet  MATH  Google Scholar 

  40. A.M. Zhou, R.K. Zhou and H.J. Haubold, The H-function: Theory and Applications. Springer, New York (2010).

    Google Scholar 

  41. R. Zhou and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.

    MathSciNet  MATH  Google Scholar 

  42. R. Zhou and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161–R208.

    MathSciNet  MATH  Google Scholar 

  43. S.I. Muslih, Solutions of a particle with fractional δ-potential in a fractional dimensional space. Int. J. Theor. Phys. 49 (2010), 2095–2104.

    MathSciNet  MATH  Google Scholar 

  44. M. Naber, Time fractional Schrödinger equation. J. Math. Phys. 45 (2004), 3339–3352.

    MathSciNet  MATH  Google Scholar 

  45. B.N. Narahari, Zhou, B.T. Yale, and J.W. Hanneken, Time fractional Schrödinger equation revisited. Adv. Math. Phys. 2013 (2013), # 290216.

    MATH  Google Scholar 

  46. J. Paneva-Konovska, Convergence of series in three parametric Mittag- Leffler functions. Math. Slovaca 64 (2014), 73–84.

    MathSciNet  MATH  Google Scholar 

  47. J. Paneva-Konovska, On the multi-index (3m-parametric) Mittag- Leffler functions, fractional calculus relations and series convergence. Cent. Eur. J. Phys. 11 (2013), 1164–1177.

    Google Scholar 

  48. J. Paneva-Konovska, Series in Mittag-Leffler functions: Inequalities and convergent theorems. Fract. Calc. Appl. Anal. 13 (2010), 403–414; at http://www.math.bas.bg/∼fcaa.

    MathSciNet  MATH  Google Scholar 

  49. T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7–15.

    MathSciNet  MATH  Google Scholar 

  50. R. Rach, On the Adomian (decomposition) method and comparisons with Picard’s method. J. Math. Anal. Appl. 128 (1987), 480–483.

    MathSciNet  MATH  Google Scholar 

  51. T. Zhou, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A: Math. Theor. 44 (2011), # 255203.

    MathSciNet  MATH  Google Scholar 

  52. T. Zhou, I. Petreska, and E.K. Lenzi, Time-dependent Schrödingerlike equation with nonlocal term. J. Math. Phys. 55 (2014), # 092105.

    MathSciNet  MATH  Google Scholar 

  53. T. Sandev, Ž. Tomovski and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise. Physica A 390 (2011), 3627–3636.

    Google Scholar 

  54. R.K. Zhou, R. Zhou and S.L. Kalla, Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics. Appl. Math. Comput. 216 (2010), 1412–1417.

    MathSciNet  MATH  Google Scholar 

  55. R.K. Zhou, R. Zhou and S.L. Kalla, Solution of spacetime fractional Schrödinger equation occurring in quantum mechanics. Frac. Calc. Appl. Anal. 13 (2010), 177–190; at http://www.math.bas.bg/∼fcaa.

    Google Scholar 

  56. H. Zhou and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), # 2455.

    Google Scholar 

  57. Ž. Zhou, T. Zhou, R. Zhou and J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A 391 (2012), 2527–2542.

    MathSciNet  Google Scholar 

  58. A.M. Wazwaz, A reliable study for extensions of the Bratu problem with boundary conditions. Math. Methods Appl. Sci. 35 (2012), 845–856.

    MathSciNet  MATH  Google Scholar 

  59. A.M. Zhou and R. Rach, Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane-Emden equations of the first and second kinds. Kybernetes 40 (2011), 1305–1318.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Delft University of Technology, NL -, 2628CD, Delft, The Netherlands

    Johan L. A. Dubbeldam

  2. Department of Mathematics, University of Rijeka, Radmile Matejcic 2, 51000, Rijeka, Croatia

    Zivorad Tomovski

  3. Faculty of Natural Sciences and Mathematics, Institute of Mathematics Saints Cyril and Methodius University, 1000, Skopje, Macedonia

    Zivorad Tomovski

  4. Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187, Dresden, Germany

    Trifce Sandev

  5. Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020, Skopje, Macedonia

    Trifce Sandev

Authors
  1. Johan L. A. Dubbeldam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zivorad Tomovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Trifce Sandev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johan L. A. Dubbeldam.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dubbeldam, J.L.A., Tomovski, Z. & Sandev, T. Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative. FCAA 18, 1179–1200 (2015). https://doi.org/10.1515/fca-2015-0068

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2015-0068

MSC 2010

Key Words and Phrases

Search

Navigation