Groups, Special Functions and Rigged Hilbert Spaces

Abstract: We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space $\mathcal H$ and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space, instead they are functionals on the dual space, $\Phi^\times$, of a rigged Hilbert space, $\Phi\subset \mathcal H \subset \Phi^\times$. As a matter of fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors $\Phi$ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can be often continuous operators on $\Phi$ with its own topology, so that they admit continuous extensions to the dual $\Phi^\times$ and, therefore, act on the elements of the continuous basis. We have investigated this formalism to various examples of interest in quantum mechanics. In particular, we have considered, $SO(2)$ and functions on the unit circle, $SU(2)$ and associated Laguerre functions, Weyl-Heisenberg group and Hermite functions, $SO(3,2)$ and spherical harmonics, $su(1,1)$ and Laguerre functions, $su(2,2)$ and algebraic Jacobi functions and, finally, $su(1,1)\oplus su(1,1)$ and Zernike functions on a circle.