An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

This paper displays the approach of the time-splitting Fourier spectral (TSFS) technique for the linear Riesz fractional Schrödinger equation (RFSE) in the semi-classical regime. The splitting technique is shown to be unconditionally stable. Further a suitable implicit finite difference discretization of second order has been manifested for the RFSE where the Riesz derivative has been discretized via an approach of fractional centered difference. Moreover the stability analysis for the implicit scheme has also been presented here via von Neumann analysis. The L²-norm and $L^{\infty }$-norm errors are calculated for $\left|u\right(x,t){|}^2$, Re$\left(u\right(x,t\left)\right)$ and Im$\left(u\right(x,t\left)\right)$ for various cases. The results obtained by the methods are further tabulated for the absolute errors for $\left|u\right(x,t){|}^2$. Furthermore the graphs are depicted showing comparison of $\left|u\right(x,t){|}^2$ by both techniques. The derivatives are taken here in the context of the Riesz fractional sense. Apart from that, the comparative study put forth in the following section via tables and graphs between the implicit second-order finite difference method (IFDM) and the TSFS method is for the purpose of investigating the efficiency of the results obtained. Moreover the stability analysis of the presented techniques manifesting their unconditional stability makes the proposed approach more competing and accurate.