Groups, Jacobi functions, and rigged Hilbert spaces

Celeghini, E.; Gadella, M.; Olmo, M. A. del

doi:10.1063/1.5138238

Cited by 4 publications

(7 citation statements)

References 56 publications

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“…Although the formalism of rigged Hilbert spaces is not new as was introduced by Gelfand [7], it has acquired more and more importance by mathematical physicists [8][9][10][11][12][13] and more recently by mathematicians [14][15][16][17][18][19][20]. Furthermore, the authors have published a series of papers in which it is shown the close connection between Lie algebras, special functions, discrete and continuous basis and rigged Hilbert spaces [22][23][24][25][26]. Somehow, the present paper could be included in this collection.…”

Section: Introductionmentioning

confidence: 84%

Hermite functions and Fourier series

Celeghini¹,

Gadella²,

Olmo³

2020

Preprint

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Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L 2 (C)) and in l 2 (Z), which are related by means of the Fourier transform and the discrete Fourier transform. This relation is unitary. Although these bases are not orthonormal the Gramm-Schmidt method permits the construction of orthonormal bases. We also have constructed ladder operators relative to the bases in both spaces, which preserve the properties of ladder operators for the one-dimensional quantum oscillator. These operators come from multiplication and differentiation operators that, when applied to periodic functions, should preserve periodicity. Finally, we have constructed riggings on these Hilbert spaces such that all these operators are continuous.

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Section: Introductionmentioning

confidence: 84%

Hermite functions and Fourier series

Celeghini¹,

Gadella²,

Olmo³

2020

Preprint

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“…Jacobi polynomials yield to the main concept of this section, the algebraic Jacobi functions [66,71], defined as…”

Section: Algebraic Jacobi Functionsmentioning

confidence: 99%

“…Then, if we choose |g = |j, m and omit the arbitrary |f ∈ Φ I,H , we have the converse relation to (71) as…”

Section: Associated Laguerre Functions On the Planementioning

confidence: 99%

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Groups, Special Functions and Rigged Hilbert Spaces

Celeghini

Gadella

Olmo

2019

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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 × . As a matter of 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 be often 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 have investigated this formalism to various examples of interest in quantum mechanics. In particular, we have considered, SO(2) and functions on the unit circle, SU (2) 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)⊕su(1, 1) and Zernike functions on a circle.

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“…In a series of previous articles [21][22][23][24][25], we have discussed the relations between discrete and continuous basis, algebras of operators, special functions and rigged Hilbert spaces (also called Gelfand triplets). We have shown that all these concepts properly convive in the framework of rigged Hilbert spaces and not on Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

Hermite Functions and Fourier Series

2021

Self Cite

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Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

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Groups, Jacobi functions, and rigged Hilbert spaces

Cited by 4 publications

References 56 publications

Hermite functions and Fourier series

Hermite functions and Fourier series

Groups, Special Functions and Rigged Hilbert Spaces

Hermite Functions and Fourier Series

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