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

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