Analysis of a third-order absorbing boundary condition for the Schrödinger equation discretized in space

Abstract

In this paper, we consider the semidiscrete problem obtained when the Schrödinger equation is discretized in space with finite differences and a third-order absorbing boundary condition specific for this discretization, which has been developed recently in the literature, is used. The well posedness of this problem is analyzed, deducing that it is weakly ill posed similarly as when absorbing boundary conditions for the continuous equation are considered. Nevertheless, we show numerically that with the semidiscrete absorbing boundary condition bigger spatial step sizes can be used, which is essential due to the weak ill posedness of the problems.