Finite two-dimensional oscillator: I. The Cartesian model

Atakishiyev, Natig M.; Pogosyan, G. S.; Vicent, Luis Edgar; Wolf, Kurt Bernardo

doi:10.1088/0305-4470/34/44/304

Cited by 69 publications

(137 citation statements)

References 18 publications

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“…. , +j} [3]. The (discrete) position wave functions have been constructed, and are given by Krawtchouk functions (normalized symmetric Krawtchouk polynomials) [3], tending to the canonical wave functions in terms of Hermite polynomials when j → ∞.…”

Section: Introductionmentioning

confidence: 99%

“…, +j} [3]. The (discrete) position wave functions have been constructed, and are given by Krawtchouk functions (normalized symmetric Krawtchouk polynomials) [3], tending to the canonical wave functions in terms of Hermite polynomials when j → ∞. In the terminology of quantum theory of angular momentum, these discrete position wave functions are just Wigner D-functions [6], and their relation to Krawtchouk polynomials was first given by Koornwinder [23].…”

Section: Introductionmentioning

confidence: 99%

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Deformed su(1,1) Algebra as a Model for Quantum Oscillators

Jafarov

Stoilova²,

Jeugt

2012

SIGMA

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Abstract. The Lie algebra su(1, 1) can be deformed by a reflection operator, in such a way that the positive discrete series representations of su(1, 1) can be extended to representations of this deformed algebra su(1, 1) γ . Just as the positive discrete series representations of su(1, 1) can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of su(1, 1) γ can be utilized to construct models of a quantum oscillator. In this case, the wave functions are expressed in terms of continuous dual Hahn polynomials. We study some properties of these wave functions, and illustrate some features in plots. We also discuss some interesting limits and special cases of the obtained oscillator models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deformed su(1,1) Algebra as a Model for Quantum Oscillators

Jafarov

Stoilova²,

Jeugt

2012

SIGMA

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“…In practice, note that (6) and (5) follow from (4) and the Jacobi-identity. So it is sufficient to verify that (4) is satisfied for the action (10), and that (10) implies (11) using (4). Note that |β, 0 is a generating vector for the representation, since…”

Section: Introductionmentioning

confidence: 99%

Discrete series representations for $\mathfrak {sl}(2|1)$, Meixner polynomials and oscillator models

Jafarov

Jeugt²

2012

J. Phys. A: Math. Theor.

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We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number β > 0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter γ. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the position operator can be continuous or infinite discrete, depending on the value of γ. We determine the position wavefunctions both in the continuous and discrete case, and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2) ⊂ sl(2|1), it can be seen why the case γ = 1 corresponds to the paraboson oscillator. Consequently, taking the values (β, γ) = (1/2, 1) in the sl(2|1) model yields the canonical oscillator.

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“…Several of such models are solved using the theory of the classical discrete polynomials [22]. Important instances of such systems are the discrete oscillators of Charlier [5], Kravchuk oscillators [6,8,10,12,14] and Meixner oscillators [5] that are related to the polynomials of Charlier, Kravchuk and Meixner, respectively, and the finite radial oscillator [9,11] related with the Hahn polynomials. For applications it is important to have recurrence relations for the discrete wave function of such systems.…”

Section: Introductionmentioning

confidence: 99%

“…Our main aim in this paper is to present a constructive approach for generating recurrence relations and ladder-type operators for some discrete system such as the discrete oscillators [2,5,6,8,9,10,11,12,13,14], discrete Calogero-Sutherland model [20], etc. The main idea is to use the connection of the wave functions with the classical discrete polynomials in a similar way as it was done in our previous paper [16] for the N -th dimensional oscillators and hydrogenlike atoms.…”

Section: Introductionmentioning

confidence: 99%