A Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schr\"odinger equation on a semi-infinite strip

TL;DR

This paper develops a stable and efficient numerical scheme combining Numerov-Crank-Nicolson and Strang splitting with transparent boundary conditions for solving the 2D Schrödinger equation on a semi-infinite strip, including practical algorithms and numerical validation.

Contribution

It introduces a novel splitting scheme with transparent boundary conditions for the Schrödinger equation, proving stability and providing an efficient FFT-based implementation.

Findings

The scheme is unconditionally stable and conservative in the L^2 sense.

Numerical results demonstrate accurate simulation of tunneling effects.

The method effectively handles general potentials with practical error control.

Abstract

We consider an initial-boundary value problem for a 2D time-dependent Schr\"odinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip is developed to implement the splitting method for general potential. Numerical results on the tunnel effect for smooth and rectangular barriers together with the practical error analysis on refining meshes are included as well.