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 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, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) ⊕ s u ( 1 , 1 ) and Zernike functions on a circle.