Accurate computation of eigenfunctions for Schrödinger operators in one dimension

A numerical method is developed to obtain sequences of functions converging to the eigenfunctions of a Schrödinger operator in the Hilbert space L2(−∞, ∞), whose norm is used to introduce the criterion of convergence in the norm and we show that it guarantees the accurate computation of expected values of a symmetric operator. The method consists in solving the Dirichlet problem associated to the eigenvalue problem in the interval [−n, n] by the Ritz method, whose convergence to both eigenvalues and eigenfunctions is guaranteed by the compactness criterion. Using the asymptotic perturbation theory in L2(−∞, ∞), we prove the convergence of both eigenvalues and eigenfunctions of the Dirichlet problem to those of the unbounded system when the interval [−n, n] is expanded. The method is applied to the harmonic oscillator, the Mitra potential, as well as to the potential V(r) = r and the Coulomb and Yukawa potentials; in each case, the convergence of eigenvalues and eigenfunctions is shown.