A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped

TL;DR

The work tackles the multidimensional time-dependent Schrödinger equation in unbounded domains by developing a high-order spatial discretization based on the Numerov operator, augmented with a Strang-type splitting in the potential and discrete transparent boundary conditions. The authors introduce a double-splitting scheme using operators $\overline{s}_N$ and $\bar{Δ}_{hN}$ to preserve favorable spectral properties for $n\ge 2$, achieving an overall accuracy of $O(\tau_{\max}^2+|h|^4)$ and unconditional $L^2$-stability on an infinite mesh. They then adapt the method to a finite mesh with a rigorously derived discrete TBC, enabling an FFT-based, dimensionally decoupled implementation into independent 1D problems with proven uniqueness and $L^2$-stability. Numerical experiments in a 2D semi-infinite strip validate the approach on tunnel-type scenarios (Pöschl-Teller barrier and rectangular well), showing minimal reflections and favorable behavior on coarse meshes. Overall, the paper provides a theoretically solid, computationally efficient framework for stable, high-order Schrödinger simulations in semi-infinite domains with practical boundary treatment.

Abstract

An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included.

Figures (5)

• Figure 1: Example A. The absolute (left) and relative (right) differences in $C$ and $L^2$ norms between the solutions of two schemes for $(J_1,J_2,M)=(800,128,1000)$ in dependence with time
• Figure 2: Example B. The absolute (left) and relative (right) differences in $C$ and $L^2$ norms between the numerical solutions for $(J_1,J_2,M)=(600,64,2400)$ and $(1200,128,4800)$ in dependence with time
• Figure 3: Example B. The modulus and the real part of the numerical solution $\Psi^m$, $m=0, 416, 616$ and $818$
• Figure 4: Example B. The modulus and the real part of the numerical solution $\Psi^m$, $m=1056, 1318, 1800$ and $2400$
• Figure 5: Example B. The total kinetic and potential energies in the cases of the well (left) and the barrier (right) in time

Theorems & Definitions (8)

• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Lemma 4.1
• proof
• Proposition 4.2
• proof