Stability and convergence analysis of artificial boundary conditions for the Schrödinger equation on a rectangular domain

Based on the semi-discrete artificial boundary condition introduced by Ji, Yang, and Antoine [Comput. Phys. Commun. 222 (2018), pp. 84–93] for two-dimensional free Schrödinger equation in a computational rectangular domain, we propose to analyze stability convergence rate with respect time of resulting full scheme. We prove that global scheme is $L^{2}$-stable accuracy second-order time, confirming then numerical results reported Ji et al. (2018).