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

doi:10.1007/s42967-025-00484-4

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 1050-1075.doi: 10.1007/s42967-025-00484-4

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

Yijin Gao¹, Songting Luo²

1. School of Economics and Finance, Shanghai International Studies University, Shanghai, 201620, China;
2. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

收稿日期:2024-04-03 修回日期:2024-12-11 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Songting Luo, Email: luos@iastate.edu E-mail:luos@iastate.edu
作者简介:Yijin Gao, Email: 2022024@shisu.edu.cn
基金资助:
Partial financial support was received from the Simons Foundation (No. 714376).

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

Yijin Gao¹, Songting Luo²

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:2024-04-03 Revised:2024-12-11 Online:2026-06-20 Published:2026-05-29
Contact: Songting Luo, Email: luos@iastate.edu E-mail:luos@iastate.edu

摘要/Abstract

摘要： 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.

关键词: Space fractional Schrödinger equation (FSE), Semiclassical approximation, Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation, Eikonal equation, Transport equation

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.

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

中图分类号:

Yijin Gao, Songting Luo. Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation II: High-Dimensional Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1050-1075.

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

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

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