Discrete transparent boundary conditions for the Schrödinger equation on circular domains

We propose transparent boundary conditions (TBCs) for the time-dependent Schrdinger 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 reflectionfree at the boundary. Then, since the discrete TBCs for the Schrdinger equation with a spatially dependent potential include a convolution with respect to 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.