Analytic Structure of Many-Body Coulombic Wave Functions

We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}$$\end{document} denote an N-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x1 = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ(1)(x) + |x1|ψ(2)(x) with ψ(1), ψ(2) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.