Appraisal of an Iterative Method for Bound States

An iterative method for determining bound-state eigenvalues and properties of the radial Schr\"odinger equation is appraised. The method stems from iterating the integral equation ψ=μ(T+1/2γ²)^-1(-Vψ), where T and V are the kinetic- and potential-energy operators. The basic theory is briefly reviewed, and calculations are performed for the Coulomb and screened-Coulomb potentials. The lowest three μ eigenvalues, together with the expected values of (γr)^-1, γr, and (γr)², are obtained from a single iterated eigenfunction sequence. Convergence is rapid for eigenvalues but slow for expected values. There is some sensitivity to the choice of the numerical integration formula. Regarded as a numerical method, this approach may be most competitive for the determination of zero-energy potential-strength eigenvalues. Its disadvantages are listed. Analytical improvements to eigenfunctions can be easier to obtain by iteration than by perturbation, and some success has been achieved. A simple example suggests that the rate of convergence of an iterated eigenfunction sequence is less than that of a related perturbation sequence unless the choice of starting function is bad.