Fourier Pseudospectral Method For Solving The Fractional Stationary Schr(?)dinger Equation

The fractional stationary Schr(?)dinger equation is one of the foundational equations of fractional quantum mechanics,which plays an important role in the study of atomic structure,molecular dynamics,quantum chemistry,nuclear physics and so on.There is a great challenge to analytically solve the ground station solution of the fractional stationary Schr¨odinger equation because of the nonlocality of the fractional Laplacian.Therefore,it is of great significance to further study the fractional stationary Schr¨odinger equation on the basis of the existing results.In this paper,we present an energy-diminishing numerical method for solving the ground state solution of the fractional stationary Schr¨odinger equation.We first construct a continuous normalized fractional gradient flow(CNFGF)and prove its energy diminishing property.Then the semidiscretization scheme is obtained by the Crank-Nicolson scheme for temporal discretization,and the full discretization scheme is developed by Fourier pseudospectral method for spatial discretization.Moreover,the time semidiscretization scheme and the full discretization scheme are proved to be energy-diminishing.Finally,the numerical experiments are given to demonstrate the efficiency of our numerical method.