Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension

We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the Schrödinger equation in one dimension. The exact nonreflecting boundary condition contains a nonlocal term which is a convolution integral in time, with a kernel proportional to 1/√t. The key observation is that this integral can be split into two parts: a local part and a history part, each of which allows for separate treatment. The local part is computed by a quadrature suited for square-root singularities. For the history part, we approximate the convolution kernel uniformly by a sum of exponentials. The integral can then be evaluated recursively. As a result, the computation of the nonreflecting boundary conditions is both accurate and efficient.