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Modern Physics Letters A

Dunkl–Schrödinger models with energy-dependent couplings: Exact solutions and confinement-like spectra

R. Boudjouraf, B. Hamil, Amar Benchikha, M. Merad

Published Aug 29, 2026• 0 citations
Publisher

Abstract

In this paper, we investigate the Dunkl–Schrödinger equation in the presence of energy-dependent potentials. A consistent formulation is developed by deriving modified expressions for the probability density, normalization condition, and scalar product, thereby ensuring the conservation of probability. Exact analytical solutions are obtained for both one- and three-dimensional systems. In the one-dimensional case, we study an energy-dependent harmonic oscillator and analyze the resulting energy spectrum for different forms of energy dependence. The analysis is then extended to three dimensions, where central potentials are considered, including the harmonic oscillator and Coulomb interactions. We show that the interplay between the Dunkl deformation and the energy dependence leads to significant modifications of the spectral structure, including nonlinear energy behavior and, in some cases, a finite number of bound states. Concrete physical realizations of such energy-dependent potentials are discussed, including semiconductors with an energy-dependent effective mass, two-dimensional electron gases in strong magnetic fields, and cold atoms in optically engineered traps.

DOI
10.1142/s0217732326502408

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In this paper, we investigate the Dunkl–Schrödinger equation in the presence of energy-dependent potentials. A consistent formulation is developed by deriving modified expressions for the probability density, normalization condition, and scalar product, thereby ensuring the conservation of probability. Exact analytical solutions are obtained for both one- and three-dimensional systems. In the one-dimensional case, we study an energy-dependent harmonic oscillator and analyze the resulting energy spectrum for different forms of energy dependence. The analysis is then extended to three dimensions, where central potentials are considered, including the harmonic oscillator and Coulomb interactions. We show that the interplay between the Dunkl deformation and the energy dependence leads to significant modifications of the spectral structure, including nonlinear energy behavior and, in some cases, a finite number of bound states. Concrete physical realizations of such energy-dependent potentials are discussed, including semiconductors with an energy-dependent effective mass, two-dimensional electron gases in strong magnetic fields, and cold atoms in optically engineered traps.
This work investigates a class of exactly solvable quantum systems described by a generalized Dunkl--Schrödinger equation with energy-dependent interactions. A two-parameter extension of the Dunkl differential operator is introduced, incorporating both the conventional reflection contribution and an additional derivative-reflection coupling. The resulting formalism leads to a modified Heisenberg algebra, from which the continuity equation, conserved probability current, and generalized normalization condition are derived for energy-dependent potentials. As an application, an energy-dependent harmonic oscillator is studied in detail. Exact analytical expressions for the eigenfunctions and energy spectrum are obtained in terms of associated Laguerre polynomials. The combined effects of the Wigner parameter, the deformation parameter, and the energy-dependence parameter are shown to produce parity splitting, nonlinear energy spectra, and nonuniform level spacing while preserving exact solvability. The thermodynamic properties of the model are then examined through the canonical partition function using the Euler--Maclaurin summation formula. Analytical expressions for the partition function, internal energy, entropy, and specific heat are derived and analyzed numerically. The results demonstrate that the generalized Dunkl deformation and energy-dependent interaction provide effective control over both the spectral and thermal characteristics of the system, extending the class of exactly solvable quantum models with reflection symmetry.
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.
In this paper, we study the Schrödinger equation with Dunkl derivative for a free particle confined in a cylindrical potential well. We consider both the finite and infinite height cases. The Dunkl formalism introduces reflection operators that modify the structure of the Hamiltonian and affect the parity of the solutions. By working in cylindrical coordinates, we obtain exact analytical expressions for the radial and axial wavefunctions in terms of Bessel functions. The energy spectrum and the solutions are classified according to the eigenvalues of the reflection operators in the three coordinates. We analyze in detail the conditions under which the wavefunctions acquire definite parity and discuss the resulting constraints on the Dunkl parameters.
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.
In this paper, we solve exactly the Schrödinger equation for the free-particle, the pseudo-harmonic oscillator and the Mie-type potential in three dimensions with the Dunkl derivative. The equations for the radial and angular parts are obtained by using spherical coordinates and separation of variables. The wave functions and the energy spectrum for these potentials are derived in an analytical way and it is shown that our results are adequately reduced to those previously reported when we remove the Dunkl derivative parameters.