On the analytic structure of the wave function for a hydrogen atom in an analytic potential

The rate of convergence of an approximate method for solving Schrödinger’s equation depends on the ability of the approximating sequence to mimic the analytic structure of the unknown exact wave function. Thus a knowledge of the analytic structure of the wave function can be of great value when approximation schemes are designed. Consider the Schrödinger equation [− 1/2 ∇2−r−1+V(r)]Ψ(r)=EΨ(r) for a hydrogen atom in a potential V(r). The general theory of elliptic partial differential equations implies that Ψ is analytic at regular points, but no general theory is available at singular points. The present paper investigates the Coulomb singular point at r=0 and shows that, if V(r)=V1(x, y, z)+rV2(x, y, z) where V1 and V2 are analytic functions of x, y, z at x=y=z=0, then the wave function has the form Ψ(r)=Ψ1(x, y, z)+rΨ2(x, y, z) where Ψ1 and Ψ2 are analytic functions of x, y, z at x=y=z=0.