Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation

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.