Groups, Special Functions and Rigged Hilbert Spaces

Axioms

Manuel Gadella¹  MarianoA. del Olmo¹  Enrico Celeghini² 

[1] Dpto de Física Teórica, Atómica y Óptica, Universidad de Valladolid, E-47005 Valladolid, Spain;Dpto di Fisica, Università di Firenze and INF-Sezione di Firenze, Sesto Fiorentino, 50019 Firenze, Italy;

关键词: rigged Hilbert spaces;    discrete and continuous bases;    special functions;    Lie algebras;    representations of Lie groups;    harmonic analysis;   

DOI  :  10.3390/axioms8030089

来源: DOAJ

【 摘 要 】

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, ${\mathsf{\Phi }}^{\times }$ , of a rigged Hilbert space, $\mathsf{\Phi }\subset \mathcal{H}\subset {\mathsf{\Phi }}^{\times }$ . 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 $\mathsf{\Phi }$ 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 $\mathsf{\Phi }$ with its own topology, so that they admit continuous extensions to the dual ${\mathsf{\Phi }}^{\times }$ 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 $SO\left(2\right)$ and functions on the unit circle, $SU\left(2\right)$ 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.

【 授权许可】

Unknown