Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Brazilian Journal of Physics

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

D. Galetti

Published Mar 1, 2003• 17 citations
Publisher

Abstract

We discuss some results from q-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 q-commutation relations associated with the q-deformed harmonic oscillator.

DOI
10.1590/s0103-97332003000100015

Related papers

On q-Deformations and Dunkl-Deformations of Harmonic Oscillators

Andreas Rung, Zentrum Mathematik

It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.

•Jan 1, 2000•0 citations

q-deformed Hermite polynomials in q-quantum mechanics

Ralf Hinterding, Julius Wess, Sektion Physik Der Ludwig-maximilians-universität +1 more

The q-special functions appear naturally in q-deformed quantum mechanics and both sides profit from this fact. Here we study the relation between the q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss: recursion formula, generating function, Christoffel-Darboux identity, orthogonality relations and the moment functional. 1 1

•Jan 1, 1998•14 citations

New q -Hermite polynomials: characterization, operators algebra and associated coherent states

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied.

Fortschritte der Physik•Jan 1, 2015•4 citations

New $q-$Hermite polynomials: characterization, operator algebra and associated coherent states

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

This paper addresses a construction of new $q-$Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem.

arXiv:1310.1280•Oct 4, 2013•1 citation

Q-extension of the linear harmonic oscillator

Mesuma K. Atakishiyeva, Natig M. Atakishiyev, Carlos Villegas‐Blas

Overlap integrals over the full real line for a family of q- extensions of the linear harmonic oscillator wave functions in quantum mechanics are evaluated. In particular, an explicit form of the squared norms for these q-wave functions is obtained. The classical Fourier-Gauss transform connects the families with different values 0 < q < 1 and q < 1 of the deformation parameter q. An explicit expansion of the q-Hermite polynomials of Rogers in terms of the ordinary Hermite polynomials emerges as a by-product.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 6, 1998•0 citations

Generalized q-deformed oscillators, q-Hermite polynomials, generalized coherent states

Burban, I. M.

The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials.

arXiv:math-ph/0607045•Jul 20, 2006•1 citation
We discuss some results from q-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 q-commutation relations associated with the q-deformed harmonic oscillator.
It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.
This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied.
New q−q-q−Hermite polynomials: characterization, operator algebra and associated coherent states
This paper addresses a construction of new q−q-q−Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem.
Overlap integrals over the full real line for a family of q- extensions of the linear harmonic oscillator wave functions in quantum mechanics are evaluated. In particular, an explicit form of the squared norms for these q-wave functions is obtained. The classical Fourier-Gauss transform connects the families with different values 0 < q < 1 and q < 1 of the deformation parameter q. An explicit expansion of the q-Hermite polynomials of Rogers in terms of the ordinary Hermite polynomials emerges as a by-product.