Multipole expansions in four-dimensional hyperspherical harmonics

TL;DR

This paper develops a method using vector differentiation to derive explicit multipole expansions in four-dimensional hyperspherical harmonics, with potential generalization to higher dimensions and explicit Clebsch-Gordan coefficients.

Contribution

It introduces a vector differentiation approach for multipole expansions in 4D hyperspherical harmonics, providing explicit formulas and generalization to higher dimensions.

Findings

Explicit multipole expansion formulas derived

Closed-form expressions for 4D Clebsch-Gordan coefficients provided

Method applicable to higher-dimensional spaces

Abstract

The technique of vector differentiation is applied to the problem of the derivation of multipole expansions in four-dimensional space. Explicit expressions for the multipole expansion of the function $r^n C_j (\hr)$ with $\vvr=\vvr_1+\vvr_2$ are given in terms of tensor products of two hyperspherical harmonics depending on the unit vectors $\hr_1$ and $\hr_2$. The multipole decomposition of the function $(\vvr_1 \cdot \vvr_2)^n$ is also derived. The proposed method can be easily generalised to the case of the space with dimensionality larger than four. Several explicit expressions for the four-dimensional Clebsch-Gordan coefficients with particular values of parameters are presented in the closed form.