Unified description of q-deformed harmonic oscillators

TL;DR

This paper presents a unified framework for describing various q-deformed harmonic oscillators, deriving explicit representations and revealing both periodic and aperiodic solution structures.

Contribution

It introduces a unified approach to q-deformed harmonic oscillators, deriving explicit operator and wave function representations, and classifying solutions by periodicity.

Findings

Unified description of Macfarlane and Dubna q-oscillators.

Explicit coordinate representations of ladder operators and wave functions.

Existence of both periodic and aperiodic solutions.

Abstract

A wide class of q-deformed harmonic oscillators including those of Macfarlane type and of Dubna type is shown to be describable in a unified way. The Hamiltonian of the oscillator is assumed to be given by a q-deformed anti-commutator of the q-deformed ladder operators. By solving $q$-difference equations, explicit coordinate representations of ladder operators and wave functions are derived, and unified parametric representations are found for $q$-Hermite functions and related formulas for the oscillators of Macfarlane and Dubna types. In addition to the well-known solutions with globally periodic structure, there exist an infinite number of solutions with globally aperiodic structure.