Exact solutions of the generalized Dunkl oscillator in the Cartesian system

In this paper, we use the generalized Dunkl derivatives instead of the standard partial derivatives in the Schrödinger equation to obtain an explicit expression of the generalized Dunkl–Schrödinger equation in 3D. It was found that this generalized Dunkl–Schrödinger equation for the 3D harmonic oscillator is exactly solvable in the Cartesian coordinates. From the relevant commutation relations, it is evident that the symmetry possessed by the original Dunkl Harmonic oscillator is broken by the generalized Dunkl derivative. Finally, we show that energy levels can be affected by considering a deformation parameter ɛ.