  Let $D\ge 1$ be an integer. In the Enright-Howe-Wallach classification list
of the unitary highest weight modules of $\widetilde{\mr{Spin}}(2, D+1)$, the
(nontrivial) Wallach representations in Case II, Case III, and the mirror of
Case III are special in the sense that they are precisely the ones that can be
realized by the Hilbert space of bound states for a generalized hydrogen atom
in dimension D. It has been shown recently that each of these special Wallach
representations can be realized as the space of L^2-sections of a canonical
hermitian bundle over the punctured ${\bb R}^D$. Here a simple algebraic
characterization of these special Wallach representations is found.
