Discrete transparent boundary conditions for the two dimensional Schrodinger equation

This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrodinger equation on a circular domain. Discrete TBCs are introduced in the numerical simulations of problems on unbounded domains in order to reduce the computa- tional domain to a finite region in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretiza- tion of such TBCs for the two-dimensional Schrodinger equation in conjunction with a conservative Crank-Nicolson-type finite difference discretization. The presented discrete TBCs yield an uncondition- ally stable numerical scheme and are completely reflection-free at the boundary. Furthermore we prove concisely the stability of the recur- rence formulas used to obtain the convolution coefficients ofthe new discrete TBC for a spatially dependent potential.