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arXiv:2212.13575 · Annals of Physics

The Dunkl oscillator on a space of nonconstant curvature: An exactly solvable quantum model with reflections

Ángel Ballesteros, Amene Najafizade, H. Panahi, HASSAN HASSANABADI, Shi‐Hai Dong

Published Nov 15, 2023• 33 citations
arXivHTMLPDF

Abstract

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions as a λ−deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of the λ−deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ−deformation and the magnetic field is explicitly illustrated.

DOI
10.1016/j.aop.2023.169543

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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.
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.
We study persistence of solvability in nonrelativistic quantum systems with positiondependent mass upon introduction of a deformation by Dunkl operators. Conditions are derived for the governing Schrödinger equation of the conventional system to admit the same solutions as in the deformed case, up to a reparametrisation of coupling constants. These conditions require the positiondependent mass or the potential of the system to have a specific form. If this is the case for a particular system, then the Schrödinger equations for its conventional version and for the Dunkl-deformed partner share solutions in the same functional form.
We present an exact, time-dependent solution for a two-dimensional Pauli oscillator deformed by Dunkl operators in the presence of an Aharonov--Bohm (AB) flux. By replacing conventional momenta with Dunkl momenta and allowing arbitrary time dependence in both, mass and frequency, we derive a deformed Pauli Hamiltonian that encodes reflection symmetries and topological gauge phases. Employing the Lewis-Riesenfeld invariant method, we derive exact expressions for the eigenvalues and spinor eigenfunctions of the system. Crucially, the AB flux imposes symmetry constraints on the Dunkl parameters of the form $ν_1 = \mp ν_2 ,linkingthereflectionsymmetry(, linking the reflection symmetry (,linkingthereflectionsymmetry(ε= \pm 1 $) to the quantization of angular momentum. These constraints modify the energy spectrum and wavefunctions of the angular operator and the invariant operator. Our framework reveals novel spectral characteristics arising from the interplay between topology and Dunkl symmetry, with potential implications for quantum simulation in engineered systems such as cold atoms and quantum dots.
Abstract We present an exact, time-dependent solution for a two-dimensional Pauli oscillator deformed by Dunkl operators in the presence of an Aharonov–Bohm (AB) flux. By replacing conventional momenta with Dunkl momenta and allowing arbitrary time dependence in both, mass and frequency, we derive a deformed Pauli Hamiltonian that encodes reflection symmetries and topological gauge phases. Employing the Lewis-Riesenfeld invariant method, we derive exact expressions for the eigenvalues and spinor eigenfunctions of the system. Crucially, the AB flux imposes symmetry constraints on the Dunkl parameters of the form ν 1 = ∓ ν 2 , linking the reflection symmetry ( ϵ = ± 1) to the quantization of angular momentum. These constraints modify the energy spectrum and wavefunctions of the angular operator and the invariant operator. Our framework reveals novel spectral characteristics arising from the interplay between topology and Dunkl symmetry, with potential implications for quantum simulation in engineered systems such as cold atoms and quantum dots.