A splitting higher order scheme with discrete transparent boundary conditions for the Schr\"odinger equation in a semi-infinite parallelepiped

TL;DR

This paper develops a high-order splitting finite-difference scheme with transparent boundary conditions for the Schrödinger equation in semi-infinite domains, ensuring stability, efficiency, and accurate numerical results for quantum tunneling problems.

Contribution

It introduces a novel higher order splitting scheme with discrete transparent boundary conditions for the Schrödinger equation in semi-infinite domains, improving spectral properties and computational efficiency.

Findings

Proves unconditional stability and L^2-conservativeness of the method.

Demonstrates effectiveness through numerical simulations of quantum tunneling.

Provides an efficient FFT-based algorithm for general potentials.

Abstract

An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schr\"odinger 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 P\"{o}schl-Teller-like potential-barrier and a rectangular potential-well are also included.