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Discrete Transparent Boundary Conditions for the Schrodinger Equation

Summary: Discrete Transparent Boundary Conditions
for the Schr¨odinger Equation
Matthias Ehrhardt, Anton Arnold
Fakult¨at f¨ur Mathematik und Informatik,
Universit¨at des Saarlandes,
Postfach 15 11 50,
D-66041 Saarbr¨ucken, Germany
ehrhardt@num.uni-sb.de, arnold@num.uni-sb.de
This paper is concerned with transparent boundary conditions for the one dimensional
time­dependent Schr¨odinger equation. They are used to restrict the original PDE prob-
lem 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 appro-
priate discretization of such transparent boundary conditions in conjunction with some
chosen discretization of the PDE (usually Crank­Nicolson finite differences in the case
of the Schr¨odinger equation). The presented discrete transparent boundary conditions
yield an unconditionally stable numerical scheme and are completely reflection­free at
the boundary.
1. Introduction
Many physical problems are described mathematically by a partial differential equation


Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien
Ehrhardt, Matthias - Institut für Mathematik, Technische Universität Berlin


Collections: Mathematics