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arXiv:1310.1280

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

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

Published Oct 4, 2013• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1310.1280

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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.
The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.
The multi-dimensional q-deformed bosonic Newton oscillator algebra with SU(d)-symmetry is considered. In this framework, we first introduce some new properties concerning the q-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of q-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the q-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.
We use the recursion relations of the continuous [Formula: see text]-Hermite polynomials and obtain the [Formula: see text]-difference realizations of the ladder operators of a [Formula: see text]-oscillator algebra in terms of the Askey–Wilson operator. For [Formula: see text]-deformed coherent states associated with a disc in the radius [Formula: see text], we obtain a compact form in [Formula: see text]-representation by using the generating function of the continuous [Formula: see text]-Hermite polynomials, too. In this way, we obtain a [Formula: see text]-difference realization for the [Formula: see text]-oscillator algebra in the finite interval [Formula: see text] as a [Formula: see text]-generalization of known differential formalism with respect to [Formula: see text] in the interval [Formula: see text] of the simple harmonic oscillator.
(q;l,λ)−(q;l,λ)-(q;l,λ)−deformed Heisenberg algebra: representations, special functions and quantization
This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements ρ(z′,z)ρ(z',z)ρ(z′,z), derivedfrom (q;l,λ)−(q;l,λ)-(q;l,λ)−deformed boson coherent states.The induced ρ(z′,z)ρ(z',z)ρ(z′,z) self-reproducing property with the associated self-reproducing kernel K(z′,z)K(z',z)K(z′,z) is computed and analyzed. An explicit construction of novel classes of generalized continuous (q;l,λ)−(q;l,λ)-(q;l,λ)−Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.