A realization fo the q-harmonic oscillator

Atakishiev, N. M.; Suslov, Sergeĭ K.

doi:10.1007/bf01016585

Cited by 35 publications

(34 citation statements)

References 11 publications

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“…continued fractions, Eulerian series, theta functions, elliptic functions, etc; see for instance [4,5]) and physics (e.g. angular momentum [6,7] and its q-analogue [8][9][10][11], the qSchrödinger equation [12] and q-harmonic oscillators [13][14][15][16][17][18][19]). Moreover, it is well known that the connection between the representation theory of quantum algebras (Clebsch-Gordan coefficients, 3j and 6j symbols) and the q-orthogonal polynomials (see [20,21,vol III,[22][23][24]), and the important role that these q-algebras play in physical applications (see for instance [26][27][28][29][30][31] and references therein).…”

Section: Introductionmentioning

confidence: 99%

The distribution of zeros of generalq-polynomials

Álvarez-Nodarse

Buendı́a

Dehesa

1997

J. Phys. A: Math. Gen.

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Abstract.A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam-Carlitz and the celebrated little and big q-Jacobi.

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Section: Introductionmentioning

confidence: 99%

The distribution of zeros of generalq-polynomials

Álvarez-Nodarse

Buendı́a

Dehesa

1997

J. Phys. A: Math. Gen.

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“…As already pointed out by some authors [27,28,29], it is possible to introduce a Õ-generalization of the harmonic oscillator algebra by considering a Hilbert space À Õ , where Õ is the parameter already discussed before, spanned by the vectors Ò , which are generated from the vaccum ¼ by the action of a raising operator Ê. In a similar way the relations hold ÄÊ ÕÊ Ä ½ (13) where Õ is a real parameter, ¼ Õ ½, or equivalently, as presented by Feinsilver [35],…”

Section: B Rogers-szegö Polynomialsmentioning

confidence: 52%

“…The technique using the factorization method [28,29,30] starts from the difference equation associated with the Õ-deformed harmonic oscillator and obtains raising and lowering operators which obey the wellknown Õ-commutation relation. An interesting aspect of this approach is that, associated with that difference equation, there appears some polynomials which can be obtained from the basic hypergeometric functions, and that generalizes the classical polynomials, namely the Jacobi, Laguerre and Hermite polynomials [31,32,33].…”

Section: Introductionmentioning

confidence: 99%

A realization of the q-deformed harmonic oscillator: rogers-Szegö and Stieltjes-Wigert polynomials

Galetti¹

2003

Braz. J. Phys.

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We discuss some results from Õ-series that can account for the foundations for the introduction of orthogonal polynomials on the circle and on the line, namely the Rogers-Szegö and Stieltjes-Wigert polynomials. These polynomials are explicitly written and their orthogonality is verified. Explicit realizations of the raising and lowering operators for these polynomials are introduced in analogy to those of the Hermite polynomials that are shown to obey the Õ-commutation relations associated with the Õ-deformed harmonic oscillator. I IntroductionThe so called Õ-deformed algebras [1,2,3,4] have been object of interest in the physics and mathematical physics literature along the last years, and a great effort has been devoted to its understanding and development [5,6,7]. The basic interest in q-deformed algebras resides in the fact that they are deformed versions of the standard Lie algebras, which are recovered as the deformation parameter Õ goes to unity. And, in this connection, since it is known that the deformed algebras encompass a set of symmetries that is richer than that of the standard Lie algebras, one is tempted to propose that Õ-deformed algebras are the appropriate tool to be used when describing physical systems symmetries which cannot be properly treated within the Lie algebras. Notwithstanding this, the direct interpretation of the deformation parameter in these cases is sometimes incomplete or even completely lacking. To mention some particularly sucessful cases in which the physical meaning of the deformation parameter is clearly established, we could cite for instance, the XXZ-model, where the ferromagnetic/antiferromagnetic nature of a spin ½ ¾ chain of length AE can be simulated through the introduction of a Õ-deformed algebra [8], or the rotational bands in deformed nuclei and molecules which can be fitted via a Õ-rotor Hamiltonian[9, 10, 11], instead of using the variable moment of inertia (VMI model). In spite of this interpretation difficulty, a solid development has emerged, from the original studies which appeared in connection with problems related to solvable statistical mechanics models [12] [23,24], and so on. The introduction of a Õ-deformed bosonic harmonic oscillator is a subject of great interest in this context and, as a tool for providing a boson realization of the quantum algebra ×Ù Õ´¾ µ, brought to light new commutation relations [25,26,27] which have been extensively discussed in the literature.In this connection, some approaches have been put forward that aim to exhibit realizations of the Õ-deformed harmonic oscillator algebra. The technique using the factorization method [28,29,30] starts from the difference equation associated with the Õ-deformed harmonic oscillator and obtains raising and lowering operators which obey the wellknown Õ-commutation relation. An interesting aspect of this approach is that, associated with that difference equation, there appears some polynomials which can be obtained from the basic hypergeometric functions, and that generalizes the classical...

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“…[3,6,7,15,21] and reference therein) that are related with the q-polynomials. For these cases only a few recurrences are known [4].…”

Section: 2mentioning

confidence: 99%