  Explicit expressions are given for the actions and radial matrix elements of
basic radial observables on multi-dimensional spaces in a continuous sequence
of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also
given for SO(N)-reduced matrix elements of basic orbital observables. These
developments make it possible to determine the matrix elements of polynomial
and a other Hamiltonians analytically, to within SO(N) Clebsch-Gordan
coefficients, and to select an optimal basis for a particular problem such that
the expansion of eigenfunctions is most rapidly convergent.
