Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 6: 2420 - 2441

DOI: https://doi.org/10.1007/s42967-024-00384-z

ORIGINAL PAPERS

Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation

  • Yijin Gao ,
  • Paul Sacks ,
  • Songting Luo
Expand
  • 1. School of Economics and Finance, Shanghai International Studies University, Shanghai 201620, China;
    2. Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Received date: 2023-09-01

  Revised date: 2024-01-09

  Online published: 2025-12-24

Supported by

Partial financial support was received from the Simons Foundation 714376.

Fold

Abstract

The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation (SE) both analytically and numerically, where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations (PDEs) that can be derived using the standard rules of standard derivatives. However, for the space fractional Schrödinger equation (FSE), the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations, because not only the problem becomes non-local, but also the rules for the standard derivatives generally do not hold for the fractional derivatives. In this work, we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) form for the space FSE based on the quantum Riesz fractional operators. We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local, the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE, and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order. We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes. Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.

Key words： Space fractional Schrödinger equation (FSE); Semiclassical approximation; Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) ansatz; Eikonal equation; Transport equation

Cite this article

Yijin Gao , Paul Sacks , Songting Luo . Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation[J]. Communications on Applied Mathematics and Computation, 2025 , 7(6) : 2420 -2441 . DOI: 10.1007/s42967-024-00384-z

References

[1] Al-Raeei, M., El-Daher, M.S.: A numerical method for fractional Schrödinger equation of Lennard-Jones potential. Phys. Lett. A 383(26), 125831 (2019)
[2] Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013)
[3] Ashyralyev, A., Hicdurmaz, B.: On the numerical solution of fractional Schrödinger differential equations with the Dirichlet condition. Int. J. Comput. Math. 89(13/14), 1927–1936 (2012)
[4] Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175(2), 487–524 (2002)
[5] Bao, W., Jin, S., Markowich, P.A.: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25(1), 27–64 (2003)
[6] BayIn, S.: On the consistency of the solutions of the space fractional Schrödinger equation. J. Math. Phys. 53(4), 042105 (2012)
[7] BayIn, S.: Definition of the Riesz derivative and its application to space fractional quantum mechanics. J. Math. Phys. 57(12), 123501 (2016)
[8] Berry, M.V., Mount, K.: Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35(1), 315 (1972)
[9] Biccari, U., Aceves, A.B.: WKB expansion for a fractional Schrödinger equation with applications to controllability (2018). https://doi.org/10.48550/ARXIV.1809.08099
[10] Bisci, G.M., Rădulescu, V.D.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial. Differ. Equ. 54, 2985–3008 (2015)
[11] Brack, M., Bhaduri, R.K.: Semiclassical Physics. CRC Press, Boca Raton (2018)
[12] Brumfiel, G.: Laser makes molecules super-cool. Nature 2010, 1 (2010)
[13] Chand, P., Hoekstra, J.: A review of the semi-classical WKB approximation and its usefulness in the study of quantum systems. In: Proceedings of the of IEEE Semiconductor Advances for Future Electronics, pp. 13–19 (2001)
[14] Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53(4), 043507 (2012)
[15] De Oliveira, E.C., Costa, F.S., Vaz, J., Jr.: The fractional Schrödinger equation for delta potentials. J. Math. Phys. 51(12), 123517 (2010)
[16] Dong, J., Xu, M.: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 48(7), 072105 (2007)
[17] Dong, J., Xu, M.: Space-time fractional Schrödinger equation with time-independent potentials. J. Math. Anal. Appl. 344(2), 1005–1017 (2008)
[18] Edeki, S., Akinlabi, G., Adeosun, S.: Analytic and numerical solutions of time-fractional linear Schrödinger equation. Commun. Math. Appl. 7(1), 1–10 (2016)
[19] Engquist, B., Runborg, O.: Computational high frequency wave propagation. Acta Numer. 12, 181–266 (2003)
[20] Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)
[21] Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082104 (2006)
[22] Herrmann, R.: The fractional symmetric rigid rotor. J. Phys. G: Nucl. Part. Phys. 34(4), 607–625 (2007)
[23] Jiang, G., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)
[24] Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
[25] Jiang, X., Qi, H., Xu, M.: Exact solutions of fractional Schrödinger-like equation with a nonlocal term. J. Math. Phys. 52(4), 042105 (2011)
[26] Kao, C., Osher, S., Qian, J.: Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations. J. Comput. Phys. 196(1), 367–391 (2004)
[27] Katori, H., Schlipf, S., Walther, H.: Anomalous dynamics of a single ion in an optical lattice. Phys. Rev. Lett. 79, 2221–2224 (1997)
[28] Keller, J.B.: Semiclassical mechanics. SIAM Rev. 27(4), 485–504 (1985)
[29] Kramer, G.J., Farragher, N.P., van Beest, B.W.H., van Santen, R.A.: Interatomic force fields for silicas, aluminophosphates, and zeolites: derivation based on ab initio calculations. Phys. Rev. B 43, 5068–5080 (1991)
[30] Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)
[31] Laskin, N.: Time fractional quantum mechanics. Chaos Solitons Fractals 102, 16–28 (2017). (Future directions in fractional calculus research and applications)
[32] Laskin, N.: Fractional Quantum Mechanics. World Scientific, Singapore (2018)
[33] Lenzi, E., Ribeiro, H., dos Santos, M., Rossato, R., Mendes, R.: Time dependent solutions for a fractional Schrödinger equation with delta potentials. J. Math. Phys. 54(8), 082107 (2013)
[34] Lim, S.C.: Fractional derivative quantum fields at positive temperature. Physica A 363(2), 269–281 (2006)
[35] Liu, X., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
[36] Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40(6), 1117–1120 (2015)
[37] Luchko, Y.: Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys. 54(1), 012111 (2013)
[38] Luo, S., Qian, J.: Factored singularities and high-order Lax-Friedrichs sweeping schemes for point-source travel times and amplitudes. J. Comput. Phys. 230(12), 4742–4755 (2011)
[39] Luo, S., Qian, J., Zhao, H.: Higher-order schemes for 3D first-arrival travel times and amplitudes. Geophysics 77(2), T47–T56 (2012)
[40] Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002)
[41] Maslov, V.P., Fedoriuk, M.V.: Semi-classical Approximation in Quantum Mechanics. D. Reidel Publishing Company, Dordrecht (1981)
[42] Muslih, S.I., Agrawal, O.P., Baleanu, D.: A fractional Schrödinger equation and its solution. Int. J. Theor. Phys. 49(8), 1746–1752 (2010)
[43] Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)
[44] Odibat, Z., Momani, S., Alawneh, A.: Analytic study on time-fractional Schrödinger equations: exact solutions by GDTM. J. Phys.: Conf. Ser. 96, 012066 (2008)
[45] Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)
[46] Pozrikidis, C.: The Fractional Laplacian. Chapman and Hall/CRC, Boca Raton (2016)
[47] Purohit, S.: Solutions of fractional partial differential equations of quantum mechanics. Adv. Appl. Math. Mech. 5(5), 639–651 (2013)
[48] Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2), 587–628 (2014)
[49] Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28, 1049–1070 (1926)
[50] Shi, H., Chen, H.: Multiple solutions for fractional Schrödinger equations. Electron. J. Differ. Equ. 25(2015), 1–11 (2015)
[51] Shu, C.-W.: High-order numerical methods for time-dependent Hamilton-Jacobi equations. In: Mathematics and Computation in Imaging Science and Information Processing. pp. 47–91. World Scientific, Singapore(2007)
[52] Tayurskii, D., Lysogorskiy, Y.: Quantum fluids in nanoporous media-effects of the confinement and fractal geometry. Chin. Sci. Bull. 56, 3617–3622 (2011)
[53] van Beest, B.W.H., Kramer, G.J., van Santen, R.A.: Force fields for silicas and aluminophosphates based on ab initio calculations. Phys. Rev. Lett. 64, 1955–1958 (1990)
[54] Wang, S., Xu, M.: Generalized fractional Schrödinger equation with space-time fractional derivatives. J. Math. Phys. 48(4), 043502 (2007)
[55] Zhang, Y., Zhao, H., Qian, J.: High-order fast sweeping methods for static Hamilton-Jacobi equations. J. Sci. Comput. 29(1), 25–56 (2006)
[56] Zhang, Y., Zhong, H., Belić, M.R., Ahmed, N., Zhang, Y., Xiao, M.: Diffraction-free beams in fractional Schrödinger equation. Sci. Rep. 6(1), 23645 (2016)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157