Improvements to the computation of eigenvalues and eigenfunctions of two-dimensional Schrödinger equations by constant perturbation based algorithms - Baeyens, Toon; Van Daele, Marnix (Journal of computational and applied mathematics, 2022)
We present important improvements and additions to a modern technique developed by Ixaru to solve the time-dependent two-dimensional Schrödinger equation with homogeneous Dirichlet boundary conditions over a rectangular domain. The algorithm, first described in Ixaru (2010), is based on the so-called Constant Perturbation technique. In this paper, we refine and extend the algorithm with important features. We focus in particular on new algorithms for the determination of the index of the eigenvalues, for the orthonormalization of eigenfunctions, for automatic step size selection and for the accurate computation of integrals. We provide the new developments with sufficient theoretical background and numerical experiments.