Unified $(q;\alpha,\beta,\gamma;\nu)$-deformation of one-parametric q-deformed oscillator algebras

TL;DR

This paper introduces a new generalized deformation of oscillator algebras, analyzes its structure, representations, and spectrum, and links it to discrete Hermite polynomials, expanding the mathematical framework of quantum oscillators.

Contribution

It defines a unified $(q;eta,eta,eta; u)$-deformed oscillator algebra and explores its properties, representations, and spectral behavior, including a special case connected to discrete Hermite polynomials.

Findings

Derived the structure function of the deformation.

Classified irreducible representations of the algebra.

Constructed a deformed oscillator with discrete spectrum linked to Hermite polynomials.

Abstract

We define a generalized $(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra and study the number of its characteristics. We describe the structure function of deformation, analyze the classification of irreducible representations and discuss the asymptotic spectrum behaviour of the Hamiltonian. For a special choice of the deformation parameters we construct the deformed oscillator with discrete spectrum of its "quantized coordinate" operator. We establish its connection with the (generalized) discrete Hermite I polynomials.