The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions

This paper is concerned with the numerical solution to the Schrödinger equation on an infinite domain. Two exact artificial boundary conditions are introduced to reduce the original problem into an initial boundary value problem with computational domain. Then, a fully discrete difference scheme is derived. The truncation errors are analyzed in detail. The unique solvability, stability and convergence with the convergence order of O(h^3/2+τ^3/2 h^-1/2) are proved by the energy method. A numerical example is given to demonstrate the accuracy and efficiency of the proposed method. As a special case, the stability and convergence of the difference scheme proposed by Baskakov and Popov in 1991 is obtained.