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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 finitedifference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strangtype 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.MSC classification: 65M06, 65M12, 35Q40

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