Quantum Mechanics and Polynomials of a Discrete Variable

Floreanini, Roberto; Letourneux, J.; Vinet, Luc

doi:10.1006/aphy.1993.1072

Cited by 30 publications

(27 citation statements)

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“…It is well known that these are related to the discrete series representation of the Lie algebra SU (1, 1) which is bounded from below [8,9,10]. It is of interest to make the connection with our model.…”

Section: Conjugation Operators and Isospectralitymentioning

confidence: 99%

An algebraic model for the multiple Meixner polynomials of the first kind

Miki

Tsujimoto

Vinet

et al. 2012

J. Phys. A: Math. Theor.

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Section: Conjugation Operators and Isospectralitymentioning

confidence: 99%

An algebraic model for the multiple Meixner polynomials of the first kind

Miki

Tsujimoto

Vinet

et al. 2012

J. Phys. A: Math. Theor.

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“…The corresponding "angular momentum" is defined by: In order to find the eigenvalues of L 0 and the coefficients c k we use the Lanczos algorithm 367 , as formulated in 368 . From eqs (36.39) and (36.40) we find…”

Section: The "Angular Momentum" Quantum Numbermentioning

confidence: 99%

Quantum groups and their applications in nuclear physics

Bonatsos

Daskaloyannis

1999

Progress in Particle and Nuclear Physics

198

203

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Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the su q (2) rotator model and its extensions, the construction of deformed exactly soluble models (u(3)⊃so (3) model, Interacting Boson Model, Moszkowski model), the 3-dimensional q-deformed harmonic oscillator and its relation to the nuclear shell model, the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to the structure of molecules and of atomic clusters, as well as an outlook are also given.

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“…It is well known that various families of orthogonal polynomials are related to the Lie algebras that we have considered so far [10,5]. We shall review these results in this section by considering different models for su(2), su(1, 1) and ho and by showing that the associated operators Y are realized within the corresponding representations as the operators of which the polynomials are eigenfunctions.…”

Section: Realizations In Terms Of Difference Operatorsmentioning

confidence: 95%

Heun algebras of Lie type

Crampé

Vinet²,

Zhedanov

2019

Proc. Amer. Math. Soc.

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We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For su(2), this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For su(1, 1), one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra ho leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the su(1, 1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.

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Quantum Mechanics and Polynomials of a Discrete Variable

Cited by 30 publications

References 0 publications

An algebraic model for the multiple Meixner polynomials of the first kind

An algebraic model for the multiple Meixner polynomials of the first kind

Quantum groups and their applications in nuclear physics

Heun algebras of Lie type

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