Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.27589 · Physica Scripta

su(1,1) Symmetry and Exact Solutions of the Dunkl-Klein-Gordon Equation in Higher Dimensions

M. Salazar-Ramírez, B. C. Lütfüoğlu, Jesús Alfredo Martínez-Nuño

quant-phSubmitted Jun 25, 2026• Announced Jun 29, 2026• 0 citations• Revised Sep 25, 2026 (v2)
v1 · Jun 25, 2026v2 · Sep 25, 2026
arXivHTMLPDF

Abstract

We investigate the $d$-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the $\mathfrak{su}(1,1)$ 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 $d$-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the $d$-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, $\mathrm{SU}(1,1)$ 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.

Journal reference
Phys. Scr. 101 (2026) 385203
DOI
10.1088/1402-4896/aea5ae

Related papers

Exact Solutions of the Dunkl--Pauli Equation in the Presence of a Magnetic Field from an su(1,1) Algebraic Approach

M. Salazar Ramírez, B. C. Lütfüoğlu, H. Bouguerne

We investigate the two-dimensional Dunkl--Pauli equation for a spin--$1/2$ particle in the presence of an external magnetic field within an algebraic framework. While previous studies have provided exact solutions of the Dunkl--Pauli system, its underlying dynamical symmetry and algebraic structure remain largely unexplored. By employing Schrödinger factorization, we construct the generators of the $\mathfrak{su}(1,1)$ algebra and establish the associated symmetry of the radial sector. The energy spectrum is derived using irreducible unitary representations, and the corresponding Sturmian radial basis is obtained analytically. Furthermore, $\mathrm{SU}(1,1)$ coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial breathing behavior. The results show that the Dunkl deformation introduces parity-dependent modifications in the spatial structure of the system while preserving its underlying algebraic dynamics.

quant-pharXiv:2609.31554•Physica Scripta•Sep 28, 2026•0 citations

SU(1,1) solution for the Dunkl–Coulomb problem in two dimensions and its coherent states

M. Salazar–Ramírez, D. Ojeda-Guillén, R. D. Mota +1 more

We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.

arXiv:1609.03627•Modern Physics Letters A•Jun 21, 2018•42 citations

Path integral approach to the Coulomb–Dunkl–Klein–Gordon problem in higher dimensions

H. Benzair, Tahar Boudjedaa, M. Merad +1 more

In this paper, we investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.

International Journal of Modern Physics A•Jan 16, 2026•1 citation

Dunkl–Klein–Gordon equation in higher dimensions

B. Hamil, B. C. Lütfüoğlu, M. Merad

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.

arXiv:2409.12655•Modern Physics Letters A•May 28, 2025•10 citations

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

R. D. Mota, D. Ojeda-Guillén

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.

math-phCross-listedarXiv:2610.37820•Oct 29, 2026

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

R. D. Mota, D. Ojeda-Guillén, M. Salazar–Ramírez +1 more

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.

arXiv:2009.06406•Modern Physics Letters A•Mar 6, 2021•34 citations
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.
We investigate the two-dimensional Dunkl--Pauli equation for a spin--1/21/21/2 particle in the presence of an external magnetic field within an algebraic framework. While previous studies have provided exact solutions of the Dunkl--Pauli system, its underlying dynamical symmetry and algebraic structure remain largely unexplored. 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 energy spectrum is derived using irreducible unitary representations, and the corresponding Sturmian radial basis is obtained analytically. 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 breathing behavior. The results show that the Dunkl deformation introduces parity-dependent modifications in the spatial structure of the system while preserving its underlying algebraic dynamics.
We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.
In this paper, we investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.
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.
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.
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.