Interacting particles in two dimensions: numerical solution of the four-dimensional Schr\"odinger equation in a hypercube

Abstract

We study numerically the Coulomb interacting two-particle stationary states of the Schr\"odinger equation, where the particles are confined in a two-dimensional infinite square well. Inside the domain the particles are subjected to a steeply increasing isotropic harmonic potential, resembling that in a nucleus. For these circumstances we have developed a fully discretized finite difference method of the Numerov-type that approximates the four-dimensional Laplace operator, and thus the whole Schr\"odinger equation, with a local truncation error of $\mathcal{O}(h^6)$, with $h$ being the uniform step size. The method is built on a 89-point central difference scheme in the four-dimensional grid. As expected from the general theorem by Keller [Num.\ Math. \textbf{7}, 412 (1965)], the error of eigenvalues so obtained are found to be the same order of magnitude which we have proved analytically as well.