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

Abstract

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...

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

“…The weight function given by (17) coincides with the Poisson Kernel (see Example 6) with r = 1/e, that is,…”

Algebraic properties of Rogers–Szegö functions: I. Applications in quantum optics

“…Moreover, Atakishiyev and Nagiyev [38] derived an important orthogonality relation on the full real line for such polynomials, and also established a special link with the SW polynomials by means of a Fourier transform. Recently, Galetti and coworkers [40] have shown didactically both the orthogonality relations for the RS and SW polynomials, as well as obtained the explicit realizations of the raising and lowering operators for each case; ‡ in addition, the authors also proposed a Wigner function related to the RS polynomials which leads us to determine a set of well-behaved marginal distribution functions with compact support for the angle and action variables. The next property to be discussed allows us to establish a connection between the finite q-Pochhammer symbol and the RS polynomials.…”

“…describing a phase distribution for a q-deformed HO, with q being a parameter that controls the distribution width, and therefore responsible for squeezing effects [40]. In order to reinforce such an argument, figure 2 shows this particular phase distribution as a function of the angular variable ϕ ∈ [−π, π] and different values of q, where the excitation degree n of the q-deformed HO is restricted into the closed interval [0, 4] -see figures 2(a)-2(e).…”

q-deformed oscillators and D-branes on conifold

“…In this section, we consider the q-oscillator structure of the wavefunction Z N (x). The q-oscillator structure for the Rogers-Szegö polynomials and the Stieltjes-Wigert polynomials, which are related to our wavefunction Z N (x) by a change of framing [15,16,18], was studied in [19,20]. From the definition Z N (x) = A N + · 1, it follows that A + acts as the raising operator…”