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A simple difference realization of the heisenberg q-algebra

Natig M. Atakishiyev, Andreas O. Frank, Kurt Bernardo Wolf

Published Jan 1, 1993• 7 citations
OpenAlex

Abstract

A realization of the Heisenberg q‐algebra whose generators are first‐order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q‐oscillator Hamiltonian are given explicitly in terms of the q−1‐Hermite polynomials. The nonuniqueness of the measure for these q‐oscillator states is also studied.

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A simple difference realization of the Heisenberg q-algebra

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A realization of the Heisenberg q-algebra whose generators are first-order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q-oscillator Hamiltonian are given explicitly in terms of the q−1-Hermite polynomials. The nonuniqueness of the measure for these q-oscillator states is also studied.

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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.

arXiv:1310.1280•Oct 4, 2013•1 citation
A realization of the Heisenberg q‐algebra whose generators are first‐order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q‐oscillator Hamiltonian are given explicitly in terms of the q−1‐Hermite polynomials. The nonuniqueness of the measure for these q‐oscillator states is also studied.
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.
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.