四维超球谐函数的多极展开
Multipole expansions in four-dimensional hyperspherical harmonics
摘要：
本文将向量微分技术应用于四维空间多极展开式的推导问题。给出了函数$r^n C_j (\hr)$与$\vvr=\vvr_1+\vvr_2$的多极展开式的显式表达式，它是由两个依赖于单位向量$\hr_1$和$\hr_2$的超球谐函数的张量积表示的.导出了函数$(\vvr_1 \cdot \vvr_2)^n$的多极分解。所提出的方法可以很容易地推广到维数大于4的空间的情况。给出了具有特定参数值的四维Clebsch-Gordan系数的几种显式封闭形式。
英文摘要：
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.
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