Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

TL;DR

This paper extends the algebraic methods for quantum superintegrable systems to anisotropic Dunkl oscillators, deriving their spectra through algebraic techniques and introducing new superintegrable Hamiltonians.

Contribution

It introduces two new superintegrable Hamiltonians involving Dunkl oscillators and develops an algebraic approach to derive their spectra.

Findings

Constructed integrals of motion for new Dunkl oscillators.

Derived energy spectra algebraically from finite-dimensional representations.

Showed spectrum division into sectors related to physical states.

Abstract

We generalise the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally $c_{\lambda}$ extended oscillator algebras with grading. We apply the construction on two-dimensional $c_{\lambda}$ oscillators. We also introduce two new superintegrable Hamiltonians that are the anisotropic Dunkl and the singular Dunkl oscillators. We construct the integrals and using this extended approach of the Daskaloyannis method with grading and we present an algebraic derivation of the energy spectrum of the two models from the finite dimensional unitary representations and show how their spectrum divides into different sectors and relates to the physical spectrum.