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

摘要

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