Exact solutions of the 2D Dunkl–Klein–Gordon equation: The Coulomb potential and the Klein–Gordon oscillator

In this paper, we begin from the Klein-Gordon ($KG$) equation in $2D$ and change the standard partial derivatives by the Dunkl derivatives to obtain the Dunkl-Klein-Gordon ($DKG$) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the $DKG$ equation. Then, we show that $DKG$ equations for the $2D$ Coulomb potential and the Klein-Gordon oscillator are exactly solvable. For each of the problems, we find the energy spectrum from an algebraic point of view by introducing suitable sets of operators which close the $su(1,1)$ algebra and use the unitary theory of representations. Also, we find analytically the energy spectrum and eigenfunctions of the $DKG$ equations for both problems. Finally, we show that when the parameters of the Dunkl derivative vanish, our results are suitably reduced to those reported in the literature for these $2D$ problems.