2018/12/11 by Jason Kaye, Leslie Greengard, Kaye, Jason +1
Engineering · Mathematics · #35Q41 (Primary) 45P05 (Secondary) #Advanced Mathematical Physics Problems #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1812.04200
openalex publication_date 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of constructing transparent boundary conditions for the time-dependent Schrödinger equation with a compactly supported binding potential and, if desired, a spatially uniform, time-dependent electromagnetic vector potential. Such conditions prevent nonphysical boundary effects from corrupting a numerical solution in a bounded computational domain. We use ideas from potential theory to build exact nonlocal conditions for arbitrary piecewise-smooth domains. These generalize the standard Dirichlet-to-Neumann and Neumann-to-Dirichlet maps known for the equation in one dimension without a vector potential. When the vector potential is included, the condition becomes non-convolutional in time. For the one-dimensional problem, we propose a simple discretization scheme and a fast algorithm to accelerate the evaluation of the boundary condition.