An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces. (English) Zbl 1078.81021
Summary: 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) irreducible representations. 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 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.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
