The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schr\"odinger equation on a semi-infinite strip

TL;DR

This paper develops a stable, efficient numerical method for solving the 2D Schrödinger equation on semi-infinite strips using splitting techniques and discrete transparent boundary conditions, validated by numerical experiments.

Contribution

It introduces a Strang-type splitting scheme with discrete TBCs for the Schrödinger equation, providing unconditional stability and an FFT-based efficient implementation.

Findings

Unconditional uniform in time L^2 stability proved.

Effective FFT-based algorithm developed for general potentials.

Numerical results demonstrate the method's accuracy and stability.

Abstract

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.