Exact transparent boundary condition for the multidimensional Schr\"odinger equation in hyperrectangular computational domain

In this paper an exact transparent boundary condition (TBC) for the multidimensional Schr\"odinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for 2D and 3D equations reported before. A new exact fully discrete (i.e. derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two non-interacting particles is considered. The proposed boundary condition is simple, robust and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schr\"odinger equation (including multi-particle problems) is required.