Groups, Special Functions and Rigged Hilbert Spaces

@article{Celeghini2019GroupsSF,
  title={Groups, Special Functions and Rigged Hilbert Spaces},
  author={Enrico Celeghini and Manuel Gadella and Mariano A. del Olmo},
  journal={Axioms},
  year={2019},
  volume={8},
  pages={89}
}
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… 

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References

SHOWING 1-10 OF 115 REFERENCES

Groups, Jacobi functions, and rigged Hilbert spaces

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in

Coherent orthogonal polynomials

Applications of rigged Hilbert spaces in quantum mechanics and signal processing

Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In

Distribution Frames and Bases

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory

Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces

A rigged Hilbert space of Hardy‐class functions: Applications to resonances

The explicit construction of a dense subspace Φ of square integrable functions on the positive half of the real line is given. This space Φ has the properties that: (1) it is endowed with a

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved

Spherical harmonics and rigged Hilbert spaces.

This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous
...