Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation with a Vector Potential

Abstract

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.

12 pages, 3 figures

References in corpus (2)

• Modeling electron dynamics coupled to continuum states in finite volumes
• Exact transparent boundary conditions for the parabolic wave equations with linear and quadratic potentials

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