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An initial-boundary value problem for the $n$-dimensional ($n\geq 2$)time-dependent Schr\"odinger equation in a semi-infinite (or infinite)parallelepiped is considered. Starting from the Numerov-Crank-Nicolsonfinite-difference scheme, we first construct higher order scheme with splittingspace averages having much better spectral properties for $n\geq 3$. Next weapply the Strang-type splitting with respect to the potential and, third,construct discrete transparent boundary conditions (TBC). For the resultingmethod, the uniqueness of solution and the unconditional uniform in time$L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing tothe splitting, an effective direct algorithm using FFT (in the coordinatedirections perpendicular to the leading axis of the parallelepiped) isapplicable for general potential. Numerical results on the 2D tunnel effect fora P\"{o}schl-Teller-like potential-barrier and a rectangular potential-well arealso included.

A splitting higher order scheme with discrete transparent boundary conditions for the Schr\"odinger equation in a semi-infinite parallelepiped