Fast and stable evaluation of the exact absorbing boundary condition for the semi-discrete linear Schrödinger equation in unbounded domains

This paper is concerned with the numerical solution of the one-dimensional semi-discrete linear Schrödinger equation in unbounded domains. In order to compute the solution on the domain of physical interest, the artificial boundary method is applied to transform the original unbounded domain problem into an initial boundary value problem on a truncated finite domain. We prove the stability of the truncated semi-discrete problem. Then, a fast algorithm is proposed to approximate the nonlocal absorbing boundary condition. The novelty of this fast algorithm is that the stability of the approximate truncated semi-discrete problem is automatically maintained. In the end, numerical examples are presented to demonstrate the performance of the proposed algorithm.