Analytic structure of many-body Coulombic wave functions

TL;DR

This paper analyzes the local analytic structure of many-electron Coulomb wave functions near coalescence points, proving they can be expressed as sums of real analytic functions multiplied by specific powers of electron-nucleus distances.

Contribution

It establishes a detailed local representation of Coulomb wave functions near coalescence points using advanced mathematical transforms and hypoellipticity, advancing understanding of their analytic properties.

Findings

Wavefunctions can be expressed as sums of analytic functions times |x| near coalescence points.

The Kustaanheimo-Stiefel transform is crucial in the proof.

Analytic hypoellipticity underpins the analytic structure results.

Abstract

We investigate the analytic structure of solutions of non-relativistic Schr"odinger equations describing Coulombic many-particle systems. We prove the following: Let psi(x) with x=(x_1,...,x_N) in R^{3N} 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 x_1=0 and the other electron coordinates do not coincide, and differ from 0, psi can be represented locally as psi(x) = psi^(1)(x) + |x_1|psi^(2)(x) with psi^(1), psi^(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.