RT info:eu-repo/semantics/article
T1 Groups, Special Functions and Rigged Hilbert Spaces
A1 Celeghini, Enrico
A1 Gadella Urquiza, Manuel
A1 Olmo Martínez, Mariano Antonio del
AB We show that Lie groups and their respective algebras, special functions and riggedHilbert spaces are complementary concepts that coexist together in a common framework andthat they are aspects of the same mathematical reality. Special functions serve as bases for infinitedimensional 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 ofthe Lie algebra are represented by unbounded self-adjoint operators. The action of these operatorson elements of continuous bases is often considered. These continuous bases do not make senseas vectors in the Hilbert space; instead, they are functionals on the dual space, Φ×, of a riggedHilbert space, Φ ⊂ H ⊂ Φ×. In fact, rigged Hilbert spaces are the structures in which both, discreteorthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topologyon it at our convenience, depending on the studied group. The generators of the Lie algebra can oftenbe continuous operators on Φ with its own topology, so that they admit continuous extensions to thedual Φ× and, therefore, act on the elements of the continuous basis. We investigate this formalism forvarious examples of interest in quantum mechanics. In particular, we consider SO(2) and functionson the unit circle, SU(2) and associated Laguerre functions, Weyl–Heisenberg group and Hermitefunctions, SO(3, 2) and spherical harmonics, su(1, 1) and Laguerre functions, su(2, 2) and algebraicJacobi functions and, finally, su(1, 1) ⊕ su(1, 1) and Zernike functions on a circle.
YR 2019
FD 2019
LK http://uvadoc.uva.es/handle/10324/40858
UL http://uvadoc.uva.es/handle/10324/40858
LA eng
NO Axioms 2019, vol. 8, 89
DS UVaDOC
RD 01-mar-2025