Exact artificial boundary conditions for the Schrödinger equation in $R ^2$

Abstract

In this paper, we propose a class of exact artificial boundary conditions for the numerical solution of the Schrödinger equation on unbounded domains in two-dimensional cases. After we introduce a circular artificial boundary, we get an initial-boundary problem on a disc enclosed by the artificial boundary which is equivalent to the original problem. Based on the Fourier series expansion and the special functions techniques, we get the exact artificial boundary condition and a series of approximating artificial boundary conditions. When the potential function is independent of the radiant angle θ, the problem can be reduced to a series of one-dimensional problems. That can reduce the computational complexity greatly. Our numerical examples show that our method gives quite good numerical solutions with no numerical reflections.