Exact Solution of the $(2+1)$-Dimensional Dunkl-Klein-Gordon Oscillator with a Uniform Magnetic Field in Non-Commutative Phase Space

Abstract

We solve the Klein-Gordon oscillator in a non-commutative phase space, generalized using Dunkl derivatives, both with and without interaction with a uniform magnetic field. For each of these cases, we begin by introducing the Dunkl derivative into the standard commutative phase space to evaluate the reflection parity symmetries. Then, using the Seiberg-Witten map, we extend the generalized system to a non-commutative phase space. In both cases, we determine the eigenfunctions and the energy spectrum exactly (which correspond to generalized Landau levels when a magnetic field is present). Furthermore, we compute the non-relativistic limit of the system and graphically analyze the behavior of the energy spectra. Our results demonstrate that the combination of the Dunkl deformation parameters, the non-commutative phase space geometry, and the magnetic field produces a parity-dependent modification of the energy levels. Finally, we verify that when the external magnetic field or the non-commutative parameters vanish, our exact solutions correctly reduce to the commutative Dunkl models established in previous works.

The paper studies a $(2+1)$-dimensional Klein–Gordon oscillator combining Dunkl derivatives, coordinate and momentum noncommutativity, and an optional uniform magnetic field. It asks how these deformations modify the oscillator spectrum and scalar Landau levels. The construction first replaces ordinary derivatives by $D_i=\partial_i+\mu_i(1-R_i)/x_i$, with $\mu_i>-1/2$, and then applies linear phase-space shifts parameterized by $\theta$ and $\eta$. The resulting operator combines a Dunkl Laplacian, a quadratic radial potential, a term linear in Dunkl angular momentum, and reflection-dependent oscillator terms. Separation in polar coordinates uses known Jacobi-polynomial angular functions; regularity and termination of the radial series give associated Laguerre functions indexed by $k\geq0$. Closed-form energy expressions are presented for joint-reflection sectors $\epsilon=R_xR_y=\pm1$, with integer or half-integer angular labels, respectively. The reported spectra exhibit deformation- and parity-dependent shifts. Magnetic-field dependence enters through three effective coefficients, allowing the field-free formulas to be reused. Algebraic limits recover previously studied commutative models, and a positive-energy expansion gives nonrelativistic expressions. Illustrative plots vary the magnetic field and deformation parameters at $k=0$ in natural units. A qualification visible in the derivation is that individual reflections are identified as nonconserved, yet subsequent spectral formulas assign their eigenvalues while also using angular-momentum eigenfunctions. Moreover, the claimed commuting set includes angular momentum despite the reflection terms generally not commuting with it. The paper does not discuss this compatibility issue or provide an explicit limitations section.