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Groups, Special Functions and Rigged Hilbert Spaces

by 1,2,†, 2,3,† and Mariano A. del Olmo 2,3,*,†
1
Dpto di Fisica, Università di Firenze and INF-Sezione di Firenze, Sesto Fiorentino, 50019 Firenze, Italy
2
Dpto de Física Teórica, Atómica y Óptica, Universidad de Valladolid, E-47005 Valladolid, Spain
3
IMUVA—Mathematical Research Institute, Universidad de Valladolid, E-47005 Valladolid, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Axioms 2019, 8(3), 89; https://doi.org/10.3390/axioms8030089
Received: 29 June 2019 / Revised: 20 July 2019 / Accepted: 21 July 2019 / Published: 27 July 2019
(This article belongs to the Special Issue Harmonic Analysis and Applications)
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. View Full-Text
Keywords: rigged Hilbert spaces; discrete and continuous bases; special functions; Lie algebras; representations of Lie groups; harmonic analysis rigged Hilbert spaces; discrete and continuous bases; special functions; Lie algebras; representations of Lie groups; harmonic analysis
MDPI and ACS Style

Celeghini, E.; Gadella, M.; del Olmo, M.A. Groups, Special Functions and Rigged Hilbert Spaces. Axioms 2019, 8, 89.

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