On a splitting higher-order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped
An initial-boundary value problem for the n -dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite parallelepiped is considered. Starting from the Numerov–Crank–Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting double-splitting method, the uniqueness of solution and the uniform in time L2-stability are proved and an error estimate is stated. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applied to implement the scheme for general potential.