Hyperspherical parametrization harmonics

We can pass from tree a to b using the suitable Clebsch-Gordan coeficient (eq. 12). The tree (c) illustrates the hyperspherical parametrization that leads to the hyperspherical harmonics Yn- Xm(, W 9) They are related to the harmonics of tree a through the Z coeficient defined in eg. (15). The connection between (b) and (c) requires a Clebsch Gordan coefficient and a phase change related to a (see eq. (14)). 

If a hyperspherical parametrization can be represented by a tree, the coordinates can be shown to form an orthogonal set. This implies that the Laplacian on the hypersphere will contain no cross terms, the corresponding Laplace equations are separable and the hyperspherical harmonics can be constructed in closed form [24]. 

Figure 5 The tree represents the hyperspherical parametrization for the components of the Jacobi vectors describing three particles in space. li,mi eind l2,m2 can be put into correspondence with j,my and l,mi of section 4, and the upper part of the graph to the ordinary spherical harmonics. ...

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

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