Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik
We consider an initial-boundary value problem for a generalized 2D time-dependent Schr\"odinger equation on a semi-infinite strip. For the Crank-Nicolson 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 uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed 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 related practical error analysis.
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http://arxiv.org/abs/1303.3471
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