Absorbing boundary conditions for the Schrödinger equation

A large number of differential equation problems which admit traveling waves have very large (typically in nite) naturally defined domains, with boundary conditions defined at the domain boundary. To be able to numerically solve these problems in smaller subdomains of the original domain, artificial boundary conditions must be defined for these subdomains. One such artifical boundary conditions which can minimize the size of such subdomains are absorbing boundary conditions. A technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves is the imposition of absorbing boundary conditions. Such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations. A general absorbing boundary condition will be developed for the Schrödinger equation with one spatial dimension, using group velocity considerations. Previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition. The well-posedness of the Initial Boundary Value Problem of the absorbing boundary condition, coupled to the interior Schrödinger equation, will also be discussed. Extension of the general absorbing boundary condition to higher spatial dimensions is demonstrated. Numerical simulations using initial single Gaussian, double Gaussian and Pseudo-delta function distributions will be given, with comparision to exact solutions, to demonstrate the reflectivity properties of various orders of the absorbing boundary condition