Generalized Deformed Oscillators and Algebras

TL;DR

This paper explores the unification of various deformed oscillator schemes, linking them to supersymmetric quantum mechanics and identifying their structures in multiple physical and mathematical systems, revealing new algebraic symmetries.

Contribution

It demonstrates the equivalence of generalized deformed oscillator schemes and their connection to supersymmetric quantum mechanics, while identifying their presence in diverse physical and mathematical models.

Findings

Unified framework for deformed oscillators established.

Connections between deformed oscillators and SUSY-QM shown.

Identification of algebraic structures in physical systems and potentials.

Abstract

The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2 supersymmetric quantum mechanics (SUSY-QM) scheme. In addition, several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to possess the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.