The Helium Wave Equation

Abstract

This paper is a sequel to the preceding one by Gronwall. In Part I it is shown that the ground state eigenfunction, if it exists, cannot have the form $\ensuremath{\psi}=\ensuremath{\Sigma}{p,k=0}^{\ensuremath{\infty}}{s}^{p+\ensuremath{\gamma}}{a}^{p, k}(\ensuremath{\beta})cosk\ensuremath{\phi}$, where $s={r}^{\frac{1}{2}}={(r_{1}^{}{}_{}{}^{2}+r_{2}^{}{}_{}{}^{2})}^{\frac{1}{2}}$, and $\ensuremath{\gamma}$ is some constant. In Part II, it is assumed that the solution of Gronwall's infinite system of ordinary differential equations (see preceding abstract) is to be found by extrapolation from a finite system. Arguments are given to show that if the wave function is finite everywhere except at the origin, then the expansion about the origin is of the form $\ensuremath{\psi}=\ensuremath{\Sigma}{k=0}^{\ensuremath{\infty}}{c}^{(k)}(s,\ensuremath{\beta},\ensuremath{\phi}){(logs)}^{k}$, where the ${c}^{(k)}$'s are ascending power series in $s$.