Local realizations of $q$-Oscillators in Quantum Mechanics

TL;DR

This paper explores local realizations of q-oscillator algebras in quantum mechanics, revealing continuous spectra and non-Fock representations linked to deformations, with an algorithm for energy level mapping.

Contribution

It introduces a general scheme for realizing q-oscillator algebra on wave functions, highlighting non-Fock representations and a method to map energy levels.

Findings

Systems exhibit continuous spectra unlike standard oscillators.

Existence of non-Fock irreducible representations related to deformation.

An algorithm for energy level mapping is provided.

Abstract

Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described.