Hyperspherical Harmonics

Abstract

This chapter is devoted to the construction of solutions to the n-dimensional Laplace equation

$$ \Delta u = \frac{{{\partial^2}u}}{{\partial x_1^2}} + ... + \frac{{{\partial^2}u}}{{\partial x_n^2}} = 0 $$

by separation of variables in spherical coordinates. An important class of special functions which naturally occur in this work is constituted by hyperspherical harmonics. In quantum mechanis these functions are used to construct basis functions in the K-harmonic method and in the translation-invariant model of shells thus enabling one to compute the fundamental physical characteristics of light nuclei; in representation theory these functions are exploited to study representations of the rotation group and motion group over an n-dimensional Euclidean space, to name just a few applications.