Space-time Legendre-Gauss-Lobatto collocation method for the two-dimensional Schrödinger equation

In this paper, we propose a novel space-time Legendre-Gauss-Lobatto collocation method for solving the time-dependent two-dimensional Schrödinger equation with nonhomogeneous boundary conditions. We first develop a new approach for systems of ordinary differential equations in the complex domain, utilizing the multi-domain Legendre-Gauss-Lobatto collocation method. We then derive the spectral rate of convergence for the proposed method in the discrete L2-norm for the semi-discrete formulation. Numerical results demonstrate that our formulation achieves exponential convergence in both space and time, thereby validating the theoretical findings.