Absorbing boundary conditions for the time-dependent Schrödinger-type equations inR3

Abstract

Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödingertype of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semi-discrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finitedifference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann (DtN) map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth order and first order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

“…In this case, the LWNs (see Equation 54) is computed using the complex amplitude in the section L z l ¼ 0 or in the section shifted on J shift from this coordinate. The comparison of the LWN k z l computed using the analytical Equation ( 24) with the LWN k z l computed using Equation ( 54) with rules ( 55) or (56) for the reference problem demonstrates a perfect coincidence (Figure 3A,B). This means that the proposed algorithm is applicable for the computation of the LWNs at the corresponding choice of the algorithm's parameters.…”

“…The choice of parameter ρ 3 strongly influences the adaptive ABCs, as confirmed by Figure 4. Here, we depict a comparison of the LWN k z l evolution computed using Equation (24) with that computed using Equation ( 54) with criterion (55) or (56) for the solution of the reference problem (Figure 4A) and that obtained by solving the problem with the adaptive ABCs using criterion (55) (Figure 4B) or (56) (Figure 4C). As seen in Figure 4, there is an optimal value of parameter ρ 3 , which is equal to 10 À6 .…”

“…[9–34] Often, the necessity of using ABCs occurs in nonlinear optics and laser physics, and these fields of modern physics are extensively studied because of their widespread laser application [35–39]. In this regard, the development of ABCs for linear and nonlinear Schrödinger equations is a meaningful task that has attracted the attention of numerous scientists [40–61].…”

A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains

“…Starting from the initial condition (3.6), Figure 4 shows the prediction by the FCNN, compared to the exact solution. The relative error for the density, the real part, the imaginary part of the [2,31,48], and the main emphasis is usually on the reflection at the boundary. The results in Figure 4 suggest that the approximation by a FCNN exhibits an absorbing property that is similar to an absorbing boundary condition.…”