We consider an initial-boundary value problem for a generalized 2D time-dependent
Schrödinger 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 L2-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.