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 / Mathematical Sciences / Pure Mathematics / Numerical and Computational Mathematics

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Funding

Simons Foundation(714376)

RIGHTS & PERMISSIONS

Shanghai University