Fourier Pseudo-spectral Methods For The Multidimensional Space-Fractional Schr(?)dinger Equation

Schr(?)dinger equation is one of basic equations in quantum mechanics,which was proposed by Austria physicist Schr(?)dinger in 1926.In recent years,because of the widespread interest in the studies of the fractional differential equations,many researchers considered the Schr(?)dinger equation with fractional order derivative,and researched its mathmatical theory and numerical methods.Based on the existing research,we further consider the numerical methods for the fractional Schr(?)dinger equation,and mainly give the time-splitting Fourier pseudo-spectral method for solving the multidimensional space-fractional Schr(?)dinger equation.Firstly,this paper briefly introduce the research background of Schr(?)dinger equation and the research status of the space-fractional Schr(?)dinger equation,and the application of Fourier spectral method in solving the partial differential equations.In the second chapter,the Fourier pseudo-spectral method for solving onedimensional space-fractional Schr(?)dinger equation is proposed,and it is extended to solve multidimensional problem,where the Fourier pseudo-spectral method is applied in the spatial discretization,and the time-splitting method is used in the time-stepping computation.In the third chapter,the mass conservation is discussed,and the stability of the algorithm is also derived.In the fourth chapter,the interpolation error estimation of the fractional Laplacian operator is given,which provides an theoretical basis for the numerical analysis of this method.Finally,the numerical experiments have taken into account the multidimensional space-fractional Schr(?)dinger equation with the harmonic potential,the hat function potential and the square well potential,respectively.In particular,the mass conservation,Hamiltonian energy deviation and the convergence behavior are shown in the numerical results.