  We consider an initial-boundary value problem for a 2D time-dependent
Schr\"odinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson
finite-difference scheme with discrete transparent boundary conditions, the
Strang-type splitting with respect to the potential is applied. For the
resulting method, the uniqueness of a solution and the uniform in time
$L^2$-stability (in particular, $L^2$-conservativeness) are proved. Due to the
splitting, an effective direct algorithm using FFT in the direction
perpendicular to the strip is developed to implement the splitting method for
general potential. Numerical results on the tunnel effect for smooth and
rectangular barriers together with the practical error analysis on refining
meshes are included as well.
