A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain (Q813203)

The authors introduce a finite-difference approximation scheme to numerically solve the one-dimensional time-dependent Schrödinger equation on an unbounded domain. Some artificial boundary conditions are introduced in order to reduce the original problem to an initial-boundary value problem on a finite-computational domain. By applying the method of reduction of order the authors implement their scheme to the reduced problem. It is proved that the proposed scheme is uniquely solvable. Moreover, it is also proved that the proposed method is unconditionally stable and convergent. A typical example is presented and the results obtained are compared with other numerical schemes.