Appraisal of an Iterative Method for Bound States

An iterative method for determining bound-state eigenvalues and properties of the radial Schrödinger equation is appraised. The method stems from iterating the integral equation $\psi =\mu {(T+\frac{1}{2}{\gamma }^2)}^{-1\mathrm{}}(-V\psi )$, 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 $\mu$ eigenvalues, together with the expected values of ${\left(\gamma \mathcal{r}\right)}^{-1\mathrm{}}$, $\gamma \mathcal{r}$, and ${\left(\gamma \mathcal{r}\right)}^2$, 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.