Multipole expansions in four-dimensional hyperspherical harmonics

doi:10.1088/0305-4470/39/12/017

Multipole expansions in four-dimensional hyperspherical harmonics

A V Meremianin 2006 J. Phys. A: Math. Gen. 39 3099-3112 doi: 10.1088/0305-4470/39/12/017

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A V Meremianin
Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden, Germany
and
REC-010, Voronezh State University, 394006, Voronezh, Russia
E-mail: avm@mpipks-dresden.mpg.de

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 $r^n C_j ( \hat{\bf r})$ with r = r₁ + r₂ are given in terms of tensor products of two hyperspherical harmonics depending on the unit vectors $\hat{\bf r}_1$ and $\hat{\bf r}_2$ . The multipole decomposition of the function (r₁ sdot r₂)ⁿ is also derived. The proposed method can be easily generalized 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.

PACS numbers: 02.40.−k, 31.15.−p, 31.15.Ja

Print publication: Issue 12 (24 March 2006)
Received 24 October 2005, in final form 2 February 2006
Published 8 March 2006

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Multipole expansions in four-dimensional hyperspherical harmonics

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