Discrete transparent boundary conditions for the Schrödinger equation

Abstract

This paper is concerned with transparent boundary conditions for the one dimensional time-dependent Schrödinger equation. They are used to restrict the original PDE problem that is posed on an unbounded domain onto a finite interval in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such transparent boundary conditions in conjunction with some chosen discretization of the PDE (usually Crank-Nicolson finite differences in the case of the Schrödinger equation). The presented discrete transparent boundary conditions yield an unconditionally stable scheme and are completely reflection-free at the boundary.