A Legendre-Galerkin Spectral Method For Solving The Schr(?)dinger Equations With Coulomb Potential

Based on the theory of intense lasers,this paper will introduce a Legendre-Galerkin spectral method(L-G spectral method)for solving the Schr(?)dinger equa-tion with Coulomb potential for atomic system describing strong laser pulse inter-actions,and tries to obtain a numerical solution with high accuracy of Schr(?)dinger equation with Coulomb potential,so as to better simulate the atomic excited state and ionization in accordance with the real situation.Legendre-Galerkin spectral method(L-G spectral method)is a more practical numerical method that belongs to Galerkin spectral method.It has the character-istics of high accuracy,high computing efficiency and better handling of variable coefficient problems,but there are few journal articles that introduce the specific application process of the method in detail.This paper introduces the realization process of Legendre-Galerkin spectral method(L-G spectral method)by solv-ing one-dimensional Poisson equation boundary value problem,two-dimensional Poisson equation boundary value problem and three-dimensional Poisson equa-tion boundary value problem.In the first place,it is assumed that the unknown function in the equation can be approximated by the expansion of the Lagrange basis function,and the the approximate expansion of the unknown function is substitute into the equation,then the Legendre-Gauss quadrature formula is used instead of the exact integral to obtain the equation about the unknown func-tion,and then use the existing technology to find the numerical solution of the equation.According to the realization of the Legendre-Galerkin spectral method(L-G spectral method)to solve the Poisson equation,it can be seen that the method is mainly aimed at the discreteness of the variables in the spatial direc-tion.With a clear understanding of the algorithm implementation process of the Legendre-Galerkin spectral method(L-G spectral method),the next step is to use this method to solve the specific one-dimensional and two-dimensional Schr(?)dinger equation with Coulomb potential initial boundary value problem,spatially,take the same discrete process as solving the Poisson equation;In the time direction,Crank-Nicolson method,iterative method,ADI method are used to discrete the time variables,and finally get the approximate information of Schr(?)dinger equa-tion with Coulomb potential by solving the unknown function equations.The algorithm implementation process and numerical results of the Legendre-Galerkin spectral method(L-G spectral method)based on the Lagrange basis function all verify the many advantages of this spectral method,indicating that this method is an effective method to study the ionization dynamics of atoms in a strong laser field.