An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces

Abstract

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.