Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.