Collective multipole expansions and the perturbation theory in the quantum three-body problem

TL;DR

This paper develops a collective multipole expansion approach for the three-body quantum problem, simplifying perturbation calculations and providing explicit formulas for Green function coefficients, with potential applications in molecular break-up processes.

Contribution

It introduces a collective multipole expansion of the three-body Green function that reduces computational complexity and derives explicit coefficient expressions for use in perturbation theory.

Findings

Reduction of matrix element dimensionality from twelve to six

Explicit formulas for multipole expansion coefficients

Dependence of S-wave coefficient on only three variables

Abstract

The perturbation theory with respect to the potential energy of three particles is considered. The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner $D$-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the $S$-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed.