The Helium Wave Equation

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 $\psi =\Sigma {p,k=0\mathrm{}}^{\infty }{s}^{p+\gamma }{a}^{p, k}\left(\beta \right)cosk\varphi$, where $s=r^{\frac{1}{2}}={(r_1^{}{}_{}{}^2+r_2^{}{}_{}{}^2)}^{\frac{1}{2}}$, and $\gamma$ 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 $\psi =\Sigma {k=0\mathrm{}}^{\infty }{c}^{\mathrm{}\left(k\right)\mathrm{}}(s,\beta ,\varphi ){\left(logs\right)}^k$, where the $c^{\mathrm{}\left(k\right)\mathrm{}}$'s are ascending power series in $s$.