Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions

We present a fast algorithm for the evaluation of exact, nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term that is a convolution integral in time. The kernel for that convolution is the inverse Laplace transform of the logarithmic derivative of a modified Bessel function, and the convolution integral can be split into two parts: a local part and a history part, which can be treated separately. The local part is easily handled by an appropriate quadrature. For the history part, we show that the convolution kernel can be well approximated by a sum of exponentials. Once such a representation is available, the convolution integrals can be evaluated recursively, reducing the cost from O (N²) work to O(N), where N is the number of time steps. The main technical development lies in the uniform rational approximation of the logarithmic derivative of the modified Bessel function K_v(√is).