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arXiv:2009.06406 · Modern Physics Letters A

Landau levels for the (2 + 1) Dunkl–Klein–Gordon oscillator

R. D. Mota, D. Ojeda-Guillén, M. Salazar–Ramírez, V. D. Granados

Published Mar 6, 2021• 34 citations
arXivHTMLPDF

Abstract

In this paper, we study the (2 + 1)-dimensional Klein–Gordon oscillator coupled to an external magnetic field, in which we change the standard partial derivatives for the Dunkl derivatives. We find the energy spectrum (Landau levels) in an algebraic way, by introducing three operators that close the su(1, 1) Lie algebra and from the theory of unitary representations. Also, we find the energy spectrum and the eigenfunctions analytically, and we show that both solutions are consistent. Finally, we demonstrate that when the magnetic field vanishes or when the parameters of the Dunkl derivatives are set to zero, our results are adequately reduced to those reported in the literature.

DOI
10.1142/s0217732321500668

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In this paper, we study the (2 + 1)-dimensional Klein–Gordon oscillator coupled to an external magnetic field, in which we change the standard partial derivatives for the Dunkl derivatives. We find the energy spectrum (Landau levels) in an algebraic way, by introducing three operators that close the su(1, 1) Lie algebra and from the theory of unitary representations. Also, we find the energy spectrum and the eigenfunctions analytically, and we show that both solutions are consistent. Finally, we demonstrate that when the magnetic field vanishes or when the parameters of the Dunkl derivatives are set to zero, our results are adequately reduced to those reported in the literature.
Exact Solution of the (2+1)(2+1)(2+1)-Dimensional Dunkl-Klein-Gordon Oscillator with a Uniform Magnetic Field in Non-Commutative Phase Space
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.
We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.
In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.
Abstract In this paper, we extend the path integral formalism for the Klein–Gordon oscillator by replacing the standard derivative with the Dunkl derivative in 1D, 2D, and even in d − dimensions. Using space-time Cartesian coordinates, we derive the propagators and solve the problems exactly. We determine the energy eigenvalues and corresponding wave functions for the spinless particle. In limiting cases, where the Dunkl derivative parameters are set to zero, our results converge appropriately to those found in the literature for these problems.
We investigate the ddd-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the su(1,1)\mathfrak{su}(1,1)su(1,1) algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the ddd-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the ddd-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, SU(1,1)\mathrm{SU}(1,1)SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.