Accurate computation of eigenfunctions for Schrödinger operators associated with Coulomb-type potentials

A numerical method is developed to obtain convergence to the eigenfunctions of the Schr\"odinger operators associated to singular potentials whose asymptotic behavior is of the Coulomb type. The method consists in solving the Dirichlet problem in a box with radius n by the Ritz method, whose convergence to the eigenfunctions in the norm of the Hilbert space L₂(0,n) is provided. Using a physical argument, we show that the solutions of the Dirichlet problem converge to those of the unbounded system in the norm of the Hilbert space L₂(0,{∞}) as n{→}{∞}. This last property guarantees the accurate computation of the expected values for a symmetric operator. The method is applied to the hydrogen atom, Yukawa potential, and Hulten potential; in each case we show the numerical convergence of the eigenfunctions, eigenvalues, and density moments.