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Licensed Unlicensed Requires Authentication Published by De Gruyter October 29, 2015

Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative

  • Johan L.A. Dubbeldam EMAIL logo , Zivorad Tomovski and Trifce Sandev

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.

References

[1] K. Abbaoui and Y. Cherruault, Convergence of Adomian’s method applied to nonlinear equations. Math. Comput. Modelling 20 (1994), 69-73.Search in Google Scholar

[2] K. Abbaoui and Y. Cherruault, New ideas for proving convergence of decomposition methods. Comput. Math. Appl. 29 (1995), 103-108.Search in Google Scholar

[3] G. Adomian, Stochastic Systems. Academic Press, New York (1983).Search in Google Scholar

[4] G. Adomian, Nonlinear Stochastic Operator Equations. Academic Press, New York (1986).10.1016/B978-0-12-044375-8.50012-5Search in Google Scholar

[5] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Publishers, Dordrecht (1988).Search in Google Scholar

[6] Q.M. Al-Mdallal, An efficient method for solving fractional Sturm- Liouville problems. Chaos, Solitons and Fractals 40 (2009), 183-189.10.1016/j.chaos.2007.07.041Search in 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), # 13799.10.1103/PhysRevB.58.13799Search in Google Scholar

[8] S.S. Bayin, Comment on “On the consistency of the solutions of the space fractional Schr¨odinger equation”. J. Math. Phys. 53 (2012), # 042105.10.1063/1.4739758Search in Google Scholar

[9] S.S. Bayin, Time fractional Schrödinger equation: Fox’s H-functions and the effective potential. J. Math. Phys. 54 (2013), # 012103.10.1063/1.4773100Search in 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), # 010602.10.1103/PhysRevLett.91.010602Search in Google Scholar PubMed

[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), # 011109.10.1103/PhysRevE.72.011109Search in Google Scholar PubMed

[12] E. Capelas de Oliveira, F.S. Costa, and J. Vaz Jr., The fractional Schr¨odinger equation for delta potentials. J. Math. Phys. 51 (2010), # 123517.10.1063/1.3525976Search in Google Scholar

[13] E. Capelas de Oliveira and J. Vaz Jr., Tunneling in fractional quantum mechanics. J. Phys. A: Math. Theor. 44 (2011), # 185303.10.1088/1751-8113/44/18/185303Search in Google Scholar

[14] M. Caputo, Elasticit`a e Dissipazione. Zanichelli, Bologna (1969).Search in Google Scholar

[15] Y. Cherruault and G. Adomian, Decomposition methods: A new proof of convergence. Math. Comput. Modelling 18 (1993), 103-106.Search in Google Scholar

[16] J. Dong, Green’s function for the time-dependent scattering problem in the fractional quantum mechanics. J. Math. Phys. 52 (2011), # 042103.10.1063/1.3571969Search in Google Scholar

[17] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York (1968).Search in Google Scholar

[18] D.K. Ferry, 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.Search in Google Scholar

[19] H.J. Haubold, A.M. Mathai, and R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011 (2011), # 298628.Search in Google Scholar

[20] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011).10.1142/8072Search in Google Scholar

[21] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).10.1142/3779Search in Google Scholar

[22] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284 (2002), 399-408.Search in Google Scholar

[23] R. Hilfer, On Fractional relaxation. Fractals 11 (2003), 251-257.Search in Google Scholar

[24] R. Hilfer, Foundations of fractional dynamics: A short account. In: Fractional Dynamics, Recent Advances, World Scientific, Singapore (2011), 209-227.10.1142/9789814340595_0009Search in Google Scholar

[25] R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B 104 (2000), 3914-3917. Search in Google Scholar

[26] A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80 (2009), 022103.10.1103/PhysRevE.80.022103Search in Google Scholar PubMed

[27] A. Iomin, Fractional-time Schr¨odinger equation: fractional dynamics on a comb. Chaos, Solitons and Fractals 44 (2011), 348-352.10.1016/j.chaos.2011.03.005Search in Google Scholar

[28] A. Iomin, L´evy flights in a box. Chaos, Solitons and Fractals 71 (2015), 73-77.10.1016/j.chaos.2014.12.010Search in Google Scholar

[29] M. Jeng, S.-L.-Y. Xu, E Hawkins, J.M. Schwarz, On the nonlocality of the fractional Schr¨odinger equation. J. Math. Phys. 51 (2010), # 062102.10.1063/1.3430552Search in Google Scholar

[30] X. Jiang, H. Qi, and M. Xu, Exact solutions of fractional Schr¨odingerlike equation with a nonlocal term. J. Math. Phys. 52 (2011), # 042105.10.1063/1.3576189Search in Google Scholar

[31] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, New York (1977).Search in Google Scholar

[32] N. Laskin, Fractional quantum mechanics and L´evy path integrals. Phys. Lett. A 268 (2000), 298-305.Search in Google Scholar

[33] N. Laskin, Fractional quantum mechanics. Phys. Rev. E 62 (2000), # 3135.10.1103/PhysRevE.62.3135Search in Google Scholar

[34] N. Laskin, Fractional Schr¨odinger equation. Phys. Rev. E 66 (2002), # 056108.10.1103/PhysRevE.66.056108Search in Google Scholar PubMed

[35] E.K. Lenzi, M.K. Lenzi, R. Rossato, and L.C.M. Filho, Solutions for diffusion equation with a nonlocal term. Acta Scientiarum. Technology 31 (2009), 81-86.Search in Google Scholar

[36] E.K. Lenzi, H.V. Ribeiro, H. Mukai, and R.S. Mendes, Continuoustime random walk as a guide to fractional Schr¨odinger equation. J. Math. Phys. 51 (2010), # 092102.10.1063/1.3491333Search in Google Scholar

[37] E.K. Lenzi, H.V. Ribeiro, M.A.F. dos Santos, R. Rossato, and R.S. Mendes, Time dependent solutions for a fractional Schrödinger equation with delta potentials. J. Math. Phys. 54 (2013), # 082107.10.1063/1.4819253Search in Google Scholar

[38] E.K. Lenzi, B.F. de Oliveira, L.R. da Silva, and L.R. Evangelista, Solutions for a Schr¨odinger equation with a nonlocal term. J. Math. Phys. 49 (2008), # 032108.10.1063/1.2842069Search in Google Scholar

[39] Y. Luchko, Fractional Schr¨odinger equation for a particle moving in a potential well. J. Math. Phys. 54 (2013), # 012111.10.1063/1.4777472Search in Google Scholar

[40] A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-function: Theory and Applications. Springer, New York (2010).10.1007/978-1-4419-0916-9Search in Google Scholar

[41] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1-77.Search in Google Scholar

[42] R. Metzler 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. 10.1088/0305-4470/37/31/R01Search in 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.Search in Google Scholar

[44] M. Naber, Time fractional Schr¨odinger equation. J. Math. Phys. 45 (2004), 3339-3352.Search in Google Scholar

[45] B.N. Narahari, Achar, B.T. Yale, and J.W. Hanneken, Time fractional Schr¨odinger equation revisited. Adv. Math. Phys. 2013 (2013), # 290216.Search in Google Scholar

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

[50] R. Rach, On the Adomian (decomposition) method and comparisons with Picard’s method. J. Math. Anal. Appl. 128 (1987), 480-483.Search in Google Scholar

[51] T. Sandev, R. Metzler and ˇZ . Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A: Math. Theor. 44 (2011), # 255203.10.1088/1751-8113/44/25/255203Search in Google Scholar

[52] T. Sandev, I. Petreska, and E.K. Lenzi, Time-dependent Schr¨odingerlike equation with nonlocal term. J. Math. Phys. 55 (2014), # 092105.10.1063/1.4894059Search in Google Scholar

[53] T. Sandev, ˇZ. Tomovski and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise. Physica A 390 (2011), 3627-3636.Search in Google Scholar

[54] R.K. Saxena, R. Saxena and S.L. Kalla, Computational solution of a fractional generalization of the Schr¨odinger equation occurring in quantum mechanics. Appl. Math. Comput. 216 (2010), 1412-1417.Search in Google Scholar

[55] R.K. Saxena, R. Saxena and S.L. Kalla, Solution of spacetime fractional Schr¨odinger equation occurring in quantum mechanics. Frac. Calc. Appl. Anal. 13 (2010), 177-190; at http://www.math.bas.bg/∼fcaa.Search in Google Scholar

[56] H. Scher and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), # 2455.10.1103/PhysRevB.12.2455Search in Google Scholar

[57] Ž. Tomovski, T. Sandev, R. Metzler and J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A 391 (2012), 2527-2542. Search in 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.Search in Google Scholar

[59] A.M. Wazwaz 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. Search in Google Scholar

Received: 2015-1-25
Revised: 2015-3-13
Published Online: 2015-10-29
Published in Print: 2015-10-1

© Diogenes Co., Sofia

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