Generalized Angular Momentum

Abstract

The generalized Laplacian operator ∆ can be written in the form:

$$ \Delta = \sum\limits_{{j = 1}}^{d} {\frac{{{{\partial }^{2}}}}{{\partial x_{j}^{2}}} = \frac{1}{{{{r}^{{d - 1}}}}}\frac{\partial }{{\partial r}}{{r}^{{d - 1}}}\frac{\partial }{{\partial r}}} - \frac{{{{ \wedge }^{2}}}}{{{{r}^{2}}}} $$

(2-1)

where Λ² is the generalized angular momentum operator, defined by

$$ {{ \wedge }^{2}} = - \sum\limits_{{i > j}}^{d} { \wedge _{{ij}}^{2}} $$

(2-2)

where

$$ {{ \wedge }_{{ij}}} \equiv {{x}_{i}}\frac{\partial }{{\partial {{x}_{j}}}} - {{x}_{j}}\frac{\partial }{{\partial {{x}_{i}}}} $$

(2-3)

We would like to show that if h_λ is an harmonic polynomial of order λ, then

$$ \left[ {{{ \wedge }^{2}} - \lambda \left( {\lambda + d - 2} \right)} \right]{{r}^{{ - \lambda }}}{{h}_{\lambda }} = 0 $$

(2-4)

In other words, we wish to show that r^−λh_λ is an eigenfunction of the generalized angular momentum operator Λ² belonging to the eigenvalue λ(λ+d−2). To show this, we first notice that

$$ {{r}^{{ - \lambda }}}{{h}_{\lambda }} \equiv {{Y}_{\lambda }}\left( \Omega \right) $$

(2-5)

is a pure angular function, independent of the hyperradius, so that

$$ \frac{\partial }{{\partial r}}{{Y}_{\lambda }}\left( \Omega \right) = 0 $$

(2-6)