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A. Arnold, M. Ehrhardt, M. Schulte, I. Sofronov:
"Discrete transparent boundary conditions for the Schrödinger equation on circular domains";
in: "ASC Report 31/2008", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2008, ISBN: 978-3-902627-01-8.

We propose transparent boundary conditions (TBCs) for the time-
dependent Schr¨odinger equation on a circular computational domain.
First we derive the two-dimensional discrete TBCs in conjunction with
a conservative Crank-Nicolson finite difference scheme. The presented
discrete initial boundary-value problem is unconditionally stable and
completely reflection-free at the boundary. Then, since the discrete
TBCs for the Schr¨odinger equation with a spatially dependent potential
include a convolution w.r.t. time with a weakly decaying kernel, we
construct approximate discrete TBCs with a kernel having the form of
a finite sum of exponentials, which can be efficiently evaluated by recursion.
In numerical tests we finally illustrate the accuracy, stability,
and efficiency of the proposed method.

two-dimensional Schr¨odinger equation, transparent boundary

http://www.asc.tuwien.ac.at/preprint/2008/asc31x2008.pdf