Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation II: High-Dimensional Problems

Abstract

The semiclassical approximation for the space fractional Schrödinger equation (FSE) in high-dimensional spaces is derived, where the phase and amplitude of the wavefunction are proved to be determined by an eikonal equation and a transport equation, respectively. The formulations extend the results of the one-dimensional (1-D) problems studied in Gao et al. (Commun Appl Math Comput, 2024. https://doi.org/10.1007/s42967-024-00384-z). Similarly, these equations reduce to the eikonal and transport equations in the semiclassical approximation for the standard Schrödinger equation with integer-order derivatives as the fractional-order approaches two, and the Hamiltonian is consistent with that in the classical Hamilton-Jacobi approach. High-order Lax-Friedrichs schemes with Runge-Kutta time integration and weighted essentially non-oscillatory finite-difference approximations are adopted to solve the eikonal and transport equations numerically for their solutions such that they can be used to approximate the wavefunction, along with numerical experiments to demonstrate the effectiveness of the semiclassical approximation.

Keywords

Space fractional Schrödinger equation (FSE) / Semiclassical approximation / Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation / Eikonal equation / Transport equation / 35C20 / 35Q40 / 65M99

References

[1] Abramowitz, M., Stegun, I.A., Romer, R.H.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. United States Department of Commerce, National Bureau of Standards, USA (1964)

[2] Al-Raeei M, El-Daher MS. A numerical method for fractional Schrödinger equation of Lennard-Jones potential. Phys. Lett. A, 2019, 383(26): Article ID: 125831

[3] 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)

[4] 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)

[5] Bao W, Jin S, Markowich PA. On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys., 2002, 175(2): 487-524

[6] 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)

[7] Bayın S. On the consistency of the solutions of the space fractional Schrödinger equation. J. Math. Phys., 2012, 53(4): Article ID: 042105

[8] Bayın S. Definition of the Riesz derivative and its application to space fractional quantum mechanics. J. Math. Phys., 2016, 57(12): Article ID: 123501

[9] Berenger J-P. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 1994, 114(2): 185-200

[10] Berry, M.V., Mount, K.: Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35(1), 315 (1972)

[11] Biccari, U., Aceves, A.B.: WKB expansion for a fractional Schrödinger equation with applications to controllability. arXiv:1809.08099 (2018)

[12] Bisci, G.M., Rădulescu, V.D.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial. Differ. Equ. 54, 2985–3008 (2015)

[13] Brumfiel, G.: Laser makes molecules super-cool. Nature (2010). https://doi.org/10.1038/news.2010.478

[14] Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507 (2012). https://doi.org/10.1063/1.3701574

[15] De Oliveira EC, Costa FS, Vaz JJr. The fractional Schrödinger equation for delta potentials. J. Math. Phys., 2010, 51(12): Article ID: 123517

[16] Dong J, Xu M. Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys., 2007, 48(7): Article ID: 072105

[17] Dong J, Xu M. Space-time fractional Schrödinger equation with time-independent potentials. J. Math. Anal. Appl., 2008, 344(2): 1005-1017

[18] Edeki S, Akinlabi G, Adeosun S. Analytic and numerical solutions of time-fractional linear Schrödinger equation. Commun. Math. Appl., 2016, 7(1): 1-10

[19] Engquist B, Runborg O. Computational high frequency wave propagation. Acta Numer., 2003, 12: 181-266

[20] Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

[21] Gao, Y., Sacks, P., Luo, S.: Numerical solutions for space fractional Schrödinger equation through semiclassical approximation. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00384-z

[22] Guo X, Xu M. Some physical applications of fractional Schrödinger equation. J. Math. Phys., 2006, 47(8): Article ID: 082104

[23] Heisenberg W. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys., 1925, 33: 879-893

[24] Herrmann R. The fractional symmetric rigid rotor. J. Phys. G Nucl. Part. Phys., 2007, 34(4): 607-625

[25] Jiang G, Peng D. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput., 2000, 21(6): 2126-2143

[26] Jiang G, Shu C-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 1996, 126(1): 202-228

[27] Jiang X, Qi H, Xu M. Exact solutions of fractional Schrödinger-like equation with a nonlocal term. J. Math. Phys., 2011, 52(4): Article ID: 042105

[28] Kao, C., Osher, S., Qian, J.: Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations. J. Comput. Phys. 196(1), 367–391 (2004)

[29] Katori H, Schlipf S, Walther H. Anomalous dynamics of a single ion in an optical lattice. Phys. Rev. Lett., 1997, 79: 2221-2224

[30] Kwaśnicki M. Ten equivalent definitions of the fractional Laplace operator. Fract. Calcul. Appl. Anal., 2017, 20(1): 7-51

[31] Laskin N. Fractional Schrödinger equation. Phys. Rev. E, 2002, 66 Article ID: 056108

[32] Laskin, N.: Time fractional quantum mechanics. Chaos Solitons Fract. 102, 16–28 (2017)

[33] Laskin, N.: Fractional Quantum Mechanics. World Scientific, Singapore (2018)

[34] 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., 2013, 54(8): Article ID: 082107

[35] Lim SC. Fractional derivative quantum fields at positive temperature. Physica A, 2006, 363(2): 269-281

[36] Liu X, Osher S, Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 1994, 115(1): 200-212

[37] Longhi S. Fractional Schrödinger equation in optics. Opt. Lett., 2015, 40(6): 1117-1120

[38] Luchko Y. Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys., 2013, 54(1): Article ID: 012111

[39] Luo S, Qian J. Factored singularities and high-order Lax-Friedrichs sweeping schemes for point-source traveltimes and amplitudes. J. Comput. Phys., 2011, 230(12): 4742-4755

[40] Luo S, Qian J, Zhao H. Higher-order schemes for 3D first-arrival traveltimes and amplitudes. Geophysics, 2012, 77(2): T47-T56

[41] Martinez AAn Introduction to Semiclassical and Microlocal Analysis, 2002New YorkSpringer

[42] Maslov, V.P., Fedoriuk, M.V.: Semi-classical Approximation in Quantum Mechanics. D. Reidel Publishing Company, Dordredcht (1981)

[43] Muleshkov, A., Nguyen, T.: Easy proof of the Jacobian for the n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n$$\end{document}-dimensional polar coordinates. Pi Mu Epsilon J. 14(4), 269–273 (2016)

[44] Muslih SI, Agrawal OP, Baleanu D. A fractional Schrödinger equation and its solution. Int. J. Theor. Phys., 2010, 49(8): 1746-1752

[45] Naber M. Time fractional Schrödinger equation. J. Math. Phys., 2004, 45(8): 3339-3352

[46] Odibat Z, Momani S, Alawneh A. Analytic study on time-fractional Schrödinger equations: exact solutions by GDTM. J. Phys. Conf. Ser., 2008, 96 Article ID: 012066

[47] Osher S, Shu C. High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal., 1991, 28(4): 907-922

[48] Pozrikidis, C.: The Fractional Laplacian. Chapman and Hall/CRC, London (2016)

[49] Purohit S. Solutions of fractional partial differential equations of quantum mechanics. Adv. Appl. Math. Mech., 2013, 5(5): 639-651

[50] Ros-Oton X, Serra J. The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal., 2014, 213(2): 587-628

[51] Schrödinger E. An undulatory theory of the mechanics of atoms and molecules. Phys. Rev., 1926, 28: 1049-1070

[52] Shi H, Chen H. Multiple solutions for fractional Schrödinger equations. Electron. J. Differ. Equ., 2015, 25(2015): 1-11

[53] 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)

[54] Tayurskii D, Lysogorskiy Y. Quantum fluids in nanoporous media-effects of the confinement and fractal geometry. Chin. Sci. Bull., 2011, 56: 3617-3622

[55] Wang S, Xu M. Generalized fractional Schrödinger equation with space-time fractional derivatives. J. Math. Phys., 2007, 48(4): Article ID: 043502

[56] Zhang, Y., Zhao, H., Qian, J.: High-order fast sweeping methods for static Hamilton-Jacobi equations. J. Sci. Comput. 29(1), 25–56 (2006)

[57] Zhang Y, Zhong H, Belić MR, Ahmed N, Zhang Y, Xiao M. Diffraction-free beams in fractional Schrödinger equation. Sci. Rep., 2016, 6(1): 23645

Funding

Simons Foundation(714376)

RIGHTS & PERMISSIONS

Shanghai University