 
Summary: DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR THE
SCHRšODINGER EQUATION ON CIRCULAR DOMAINS
ANTON ARNOLD , MATTHIAS EHRHARDT , MAIKE SCHULTE §, AND IVAN
SOFRONOV ¶
Abstract. We propose transparent boundary conditions (TBCs) for the timedependent
Schršodinger equation on a circular computational domain. First we derive the twodimensional
discrete TBCs in conjunction with a conservative CrankNicolson finite difference scheme. The pre
sented discrete initial boundaryvalue problem is unconditionally stable and completely reflection
free at the boundary. Then, since the discrete TBCs for the Schršodinger 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.
As a byproduct we also present a new formulation of discrete TBCs for the 1D Schršodinger
equation, with convolution coefficients that have better decay properties than those from the litera
ture.
Key words. twodimensional Schršodinger equation, transparent boundary conditions, discrete
convolution, sum of exponentials, PadŽe approximations, finite difference schemes
subject classifications. 65M12, 35Q40, 45K05
1. Introduction
