We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the SchrSdinger equation in one dimension. The exact nonrefleeting boundary condition contains a nonloeal term which is a convolution integral in time, with a kernel proportional to 1/v~. The key observation is that this integral can be split into two parts: a… (More)
A second kind integral equation formulation is presented for the Dirichlet problem for the Laplace equation in two dimensions, with the boundary conditions specified on a collection of open curves. The performance of the obtained apparatus is illustrated with several numerical examples. The formulation is a simplification of the equation previously… (More)
A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel.… (More)
We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel U (x) and a density function ρ(x) = |ψ(x)| 2 for some complex-valued wave function ψ(x), permitting the formal use of Fourier methods. These are… (More)
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 which is a convolution integral in time. The kernel for that convolution is the… (More)
The jump relations of the quadruple layer potential on a regular surface in three dimensions are derived. The jumps are shown to be proportional to the product of the density of the potential and the mean curvature of the underlying surface.
We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schrödinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in… (More)