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 computation. This is the second version of the manuscript, correcting an important omission: namely the paper [1], which derived exact transparent boundary conditions from Green's representation theorem, as we do in Section 2 below. We have made minor changes to the presentation, emphasizing that the novelty of the present work is in the derivation of a fast algorithm which makes the Green's function-based approach practical.

A fast, high-order numerical method for the simulation of single-excitation states in quantum optics

“…This leads to an algorithm which, for a given accuracy, has a computational cost scaling as O N 2 in the number N of time steps, and a memory requirement scaling as O (N ). This is a typical challenge associated with the application of Volterra integral operators, and several techniques have been proposed to address it, particularly in the context of solving Volterra integral equations [12][13][14][15][16][17][18] and applying Volterra integral operators corresponding to nonlocal transparent boundary conditions [19][20][21][22][23][24]. We will make use of one such approach -the sum of exponentials approximation method -to obtain a high-order accurate numerical method with O (N log N ) computational complexity and O (log N ) memory complexity.…”

A fast time domain solver for the equilibrium Dyson equation

“…The problem of efficient Volterra history integration is well known in the scientific computing literature, and several techniques have been proposed, particularly in the context of solving Volterra integral equations and computing nonlocal transparent boundary conditions. We mention fast Fourier transform (FFT)-based methods [27], methods based on sum-of-exponentials projections of the history [28][29][30][31][32][33][34][35], convolution quadrature [36,37], and hierarchical low-rank or butterfly matrix compression [22,38,39]. Of the methods mentioned above, several have been applied to the efficient numerical solution of convolutional nonlinear Volterra integral equations [27,37,38].…”

Computability of magnetic Schrödinger and Hartree equations on unbounded domains

“…Most of the literature dealing with numerical approximation schemes on unbounded domains, has been concerned with the existence of transparent or absorbing boundary conditions, see e.g. [KG18,JG07] and references therein, for electro-magnetic time-dependent Schrödinger equations. In [KG18] spatial restrictions are imposed on the electromagnetic vector potential, to avoid undesired boundary effects.…”

“…[KG18,JG07] and references therein, for electro-magnetic time-dependent Schrödinger equations. In [KG18] spatial restrictions are imposed on the electromagnetic vector potential, to avoid undesired boundary effects. In this work, we circumvent such undesired restrictions by studying a somewhat different question: Can we forget about the unboundedness of the domain and just restrict the dynamics to a bounded domain whose size is uniform in the input parameters (initial state, potentials, control) of our problem?…”