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

Angel Ballesteros 1<EMAIL_ADDRESS>Amene Najafizade2<EMAIL_ADDRESS>Hossein Panahi2<EMAIL_ADDRESS>Hassan Hassanabadi3
<EMAIL_ADDRESS>Shi-Hai Dong 4,5<EMAIL_ADDRESS>1Departamento
de Física, Universidad de Burgos, 09001 Burgos, Spain  2Department of Physics,
University of Guilan, Rasht 51335-1914, Iran  3Faculty of Physics, Shahrood
University of Technology, Shahrood, Iran  4Huzhou University, Huzhou, 313000,
P. R. China  5Laboratorio de Información Cuántica, CIDETEC, Instituto
Politécnico Nacional, UPALM, CDMX07700, Mexico

###### Abstract

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions,
which is defined as a $\lambda-$deformation of the N-dimensional Dunkl
oscillator. This deformation can be interpreted either as the introduction of
an non-constant curvature related with $\lambda$ on the underlying space or,
equivalently, as a Dunkl oscillator with a position-dependent mass function.
This new quantum model is shown to be exactly solvable in arbitrary dimension
N, and its eigenvalues and eigenfunctions are explicitly presented. Moreover,
it is shown that in the two-dimensional case both the Darboux III and the
Dunkl oscillators can be separately coupled with a constant magnetic field,
thus giving rise to two new exactly solvable quantum systems in which the
effect of a position-dependent mass and of the Dunkl derivatives on the
structure of the Landau levels can be explicitly studied. Finally, the full 2D
Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to
define also an exactly solvable Hamiltonian, where the interplay between the
$\lambda$-deformation and the magnetic field is explicitly illustrated.

nonconstant curvature, Dunkl-Darboux III oscillator, deformation, Dunkl
derivative, exact solutions, Landau levels.

###### pacs:

02\. 30. Jr, 03. 65. -w, 03. 65. Ge

###### Contents

  1. I Introduction
  2. II The Darboux III oscillator
    1. II.1 The one-dimensional Darboux III oscillator
    2. II.2 The N-dimensional Darboux III oscillator
    3. II.3 The 2D Darboux III oscillator coupled to a constant magnetic field
  3. III The Dunkl oscillator
    1. III.1 The one-dimensional Dunkl oscillator
    2. III.2 The ND Dunkl oscillator
    3. III.3 The 2D Dunkl oscillator coupled to a constant magnetic field
      1. III.3.1 Case $R_{x}=R_{y}$
      2. III.3.2 Case $R_{x}=-R_{y}$
  4. IV The Dunkl-Darboux III oscillator
    1. IV.1 The one-dimensional model
    2. IV.2 The ND Dunkl-Darboux III oscillator
    3. IV.3 The Dunkl-Darboux III oscillator coupled to a constant magnetic field
      1. IV.3.1 Case $R_{x}=R_{y}$
      2. IV.3.2 Case $R_{x}=-R_{y}$
  5. V Conclusions
  6. A N-dimensional hyperspherical coordinates and Dunkl derivatives

##  I Introduction

The isotropic harmonic oscillator on the $N$-dimensional Euclidean space
constitutes the paradigm of maximally superintegrable (classical and quantum)
Hamiltonian systems, whose superintegrability is deeply connected with the
existence of a maximal set of symmetries for the Hamiltonian. As it is well-
known, when the oscillator potential (including also centrifugal terms) is
suitably defined on N-dimensional spaces with constant curvature, the
superintegrability of the system is preserved, together with the
dimensionality of the algebra of symmetries (see for instance FMSUW65 ; Ev90b
; Ev91 ; 10 ; 20 ; RS ; 21 ; BH07  and references therein).

Interestingly enough, a superintegrable version of the isotropic oscillator on
a N-dimensional (ND) space with nonconstant curvature was found in physicaD ;
PLA  (see also cq0 ; cq1 ; cq2 ; cq3 ; latiniJPCS ; bert1 ; cq6 ; cq7  and
references therein). This space is the so-called ND Darboux III space, a
conformally flat space with metric

$\mathrm{d}\mathbf{s}^{2}=(1+\lambda\mathbf{x}^{2})\mathrm{d}\mathbf{x}^{2},$
(1)

and whose (negative) nonconstant scalar curvature is given by

$\mathcal{R}=-\lambda\frac{(N-1)(2N+3(N-2)\lambda\mathbf{x}^{2})}{(1+\lambda\mathbf{x}^{2})^{3}}.$
(2)

As it was shown in , the maximally superintegrable oscillator on this space is
defined by the Hamiltonian

$H(\mathbf{x},\mathbf{p})=\frac{1}{2(1+\lambda\mathbf{x}^{2})}\,(\mathbf{p}^{2}+\omega^{2}\mathbf{x}^{2})\,,$
(3)

with real parameters $\lambda>0$ and $\omega\geq 0$ and where
$\mathbf{x},\mathbf{p}$ are conjugate coordinates and momenta. Such a curved
space is just the ND spherically symmetric generalization of the Darboux
surface of type III koen ; Kal . Indeed, the Hamiltonian (3) is a genuine
$\lambda-$deformation of the well-known flat ND isotropic harmonic oscillator
with frequency $\omega$, which is straightforwardly obtained in the limit
$\lambda\to 0$:

$\mathcal{H}_{0}=\frac{1}{2}(\mathbf{p}^{2}+\omega^{2}\mathbf{x}^{2}),\qquad\mathrm{d}\mathbf{s}^{2}=\mathrm{d}\mathbf{x}^{2},\qquad\mathcal{R}=0.$
(4)

It is worth stressing that the Darboux III oscillator can be also interpreted
as a nonlinear oscillator (yet defined on the ND Euclidean space) but whose
mass $m$ is position-dependent in the form
$M(\mathbf{x})=1+\lambda\,\mathbf{x}^{2}$.

A further type of symmetry-preserving generalization of the isotropic
oscillator is given by the so-called Dunkl oscillator, where usual derivatives
are replaced by the Dunkl ones. The Dunkl operator introduced by Yang allows
the definition of a type of deformed oscillator yang , which includes of
reflection symmetries wigner , which allow the definition of new quantum
integrable models integ ; integ1 ; integ2 . Several authors have generalized
several integrable Hamiltonians by making use of the Dunkl operator, including
Dunkl-Dirac dirac1 ; dirac2 , Dunkl-Klein-Gordon gordon1 ; gordon2 , Dunkl-
pseudo harmonic oscillators pse , and Dunkl-Duffin-Kemmer-Petiau oscillators
dkp . Recently, some of the authors have studied the symmetries defining the
Schwinger-Dunkl algebra involving the Dunkl derivatives related to the -1
orthogonal polynomials of the Bannai-Ito algebra ito1 ; ito2 ; ito3 ; ito4 .
Moreover, in Refs. two ; three ; 2d  the two and three-dimensional curved
Dunkl oscillators (associated to the so-called Jordan-Schwinger-Dunkl algebra)
were studied, and the superintegrability of all these models was demonstrated.

The aim of this paper is the introduction of the Dunkl-Darboux III oscillator
Hamiltonian, in which the nonlinear deformation arising in the Darboux III
oscillator is supplemented by the additional reflection symmetries associated
to Dunkl operators. By making use of all the abovementioned techniques, this
new quantum model will be shown to be exactly solvable in the generic
N-dimensional case, and its associated wavefunctions and energy eigenvalues
will be explicitly presented. In particular, special attention will be payed
to the $N=2$ case, since for that dimensionality we will show that a constant
perpendicular magnetic field can be also introduced, thus giving rise to a
biparametric family of exactly solvable models which provide the Dunkl-Darboux
III generalization of the Landau levels.

This paper is organized as follows. In Section 2, the Darboux III oscillator
is reviewed in order to provide the necessary background needed for the rest
of the paper. Also, the new quantum Hamiltonian for the 2D Darboux III
oscillator in the presence of a constant magnetic field is introduced and
fully solved. In particular, the modifications introduced in the Landau levels
by the nonlinear deformation $\lambda$ and by the magnetic field $B$ ara
analysed in detail. The well-known ND Dunkl oscillator is revisited in Section
3, and the new quantum model defining the 2D Dunkl oscillator in the presence
of a magnetic field is also introduced and completely solved. Finally in
Section 4 the ND Dunkl-Darboux III oscillator is defined and solved by making
use of all the algebraic machinery presented in the two previous sections.
Again, for the 2D case a constant magnetic field can be also included in the
model, where it is worth noticing that the Dunkl analogue of the angular
momentum operator allows for the separation of variables in terms of polar
coordinates, and therefore leads to the explicit solution of the system.
Finally, in Section 5, we give some concluding remarks.

##  II The Darboux III oscillator

###  II.1 The one-dimensional Darboux III oscillator

In order to study the quantum systems associated to the one-dimensional
version of the classical Hamiltonian (3), let us consider the standard
definitions for quantum position $\hat{x}$ and momentum $\hat{p}_{x}$
operators, with Lie brackets and differential representation given by

$\hat{x}\psi(x)=x\psi(x),\qquad\hat{p}_{x}\psi(x)=-i\hbar\frac{\partial\psi(x)}{\partial
x},\qquad[\hat{x},\hat{p}_{x}]=i\hbar.$ (5)

Hence, the well-known quantum version of the Hamiltonian (3) is given by PLA ;
cq3

$\hat{\mathcal{H}}_{D^{\prime}}=\frac{1}{(1+\lambda
x^{2})}\left[\frac{1}{2}\hat{p}_{x}^{2}+\frac{1}{2}\omega^{2}\hat{x}^{2}\right]=\frac{1}{(1+\lambda
x^{2})}\left[-\frac{\hbar^{2}}{2}\partial_{x}^{2}+\frac{1}{2}\omega^{2}x^{2}\right],$
(6)

which is formally self-adjoint on the $L^{2}(\mathbb{R},\mathrm{d}x)$ Hilbert
space defined by the scalar product PLA ; cq3

$\langle\psi|\phi\rangle_{\lambda}=\int\overline{\psi(x)}\phi(x)(1+\lambda
x^{2})\mathrm{d}x.$ (7)

The corresponding Schrödinger equation can be written as
$\hat{\mathcal{H}}_{D^{\prime}}\psi(x)=E\psi(x)$ is

$\frac{1}{2}\left[-\hbar^{2}\partial_{x}^{2}+\left(\omega^{2}-2\lambda
E\right)x^{2}\right]\psi(x)=E\psi(x),$ (8)

which defines a quantum oscillator with an energy-dependent frequency given by

$\Omega(E)=\sqrt{\omega^{2}-2\lambda
E}\qquad\mathrm{whenever}\quad\omega^{2}>2\lambda E\,,$ (9)

and whose eigenvalues can be obtained by solving the equation

$E=\hbar\Omega\left(n+\frac{1}{2}\right)\,.$ (10)

Therefore, we have an exactly solvable one-dimensional quantum model whose
explicit eigenvalues come from solving the equation (10), and with
eigenfunctions $\psi_{n}(x)$ are given in analogy with the harmonic oscillator
wavefunctions. In this way we obtain that PLA ; cq3

$\displaystyle{E}_{n}=-\hbar^{2}\lambda\left(n+\frac{1}{2}\right)^{2}+\hbar\left(n+\frac{1}{2}\right)\sqrt{\lambda^{2}\hbar^{2}\left(n+\frac{1}{2}\right)^{2}+\omega^{2}},\qquad
n=0,1,2,\dots$ (11)
$\displaystyle\psi_{n}(x)=A_{n}\left(\frac{\beta^{2}}{\pi}\right)^{\frac{1}{4}}\exp\left[-\beta^{2}x^{2}/2\right]H_{n}(\beta
x),\qquad\beta=\sqrt{\frac{\Omega}{\hbar}},$ (12)

where $H_{n}(\beta x)$ is the $n$-th Hermite polynomial and $A_{n}$ is a
normalization constant. Note that the real parameter $\lambda$ can be
interpreted as a ”deformation” parameter governing the nonlinear behaviour of
$\hat{\mathcal{H}}_{D^{\prime}}$, and this parameter is deeply related to the
variable curvature of the underling Darboux space. The case $\lambda=0$ leads
to the well-known results for the harmonic oscillator Hamiltonian
$\hat{\mathcal{H}}=-\frac{\hbar^{2}}{2}\partial_{x}^{2}+\frac{1}{2}\omega^{2}x^{2}$
with frequency $\omega$, which indeed does not depend on the energy $E$.

###  II.2 The N-dimensional Darboux III oscillator

In the N-dimensional generalization of the Darboux III system physicaD ; PLA ;
cq3 , the quantum position and momentum operators
$(\hat{\mathbf{x}},\hat{\mathbf{p}})$ are defined by

$\hat{x}_{i}=x_{i},\qquad\hat{p}_{i}=-i\hbar\frac{\partial}{\partial
x_{i}},\qquad[\hat{x}_{i},\hat{p}_{i}]=i\hbar\delta_{ij}\,.$ (13)

Thus one can use the standard notation

$\nabla=\left(\frac{\partial}{\partial x_{1}},\dots,\frac{\partial}{\partial
x_{N}}\right),\qquad\Delta=\nabla^{2}=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\dots+\frac{\partial^{2}}{\partial x_{N}^{2}}\,,$ (14)

and from the point of view of the so-called ”Schrödinger” quantization cq3 ,
the quantum Hamiltonian is given by

$\hat{H}_{D^{\prime}}=\frac{1}{(1+\lambda\hat{\mathbf{x}}^{2})}\left[\frac{1}{2}\hat{\mathbf{p}}^{2}+\frac{1}{2}\omega^{2}\hat{\mathbf{x}}^{2}\right]=\frac{1}{(1+\lambda\mathbf{x}^{2})}\left[-\frac{\hbar^{2}}{2}\nabla^{2}+\omega^{2}\mathbf{x}^{2}\right]\,.$
(15)

This quantization prescription preserves the maximal superintegrability of the
system in a straightforward way due to the immediate quantum transcription of
the $(2N-1)$ classical integrals of the motion, which are given by the
operators PLA ; cq3

$\hat{I}_{i}=\hat{p}_{i}^{2}-2\lambda\,\hat{q}_{i}^{2}{\hat{H}}_{D^{\prime}}+\omega^{2}\hat{q}_{i}^{2},\qquad
i=1,\dots,N\,,$ (16)

and therefore ${\hat{H}}_{D^{\prime}}=\frac{1}{2}\sum_{i=1}^{N}\hat{I}_{i}$.

By taking into account a factorized wave function together with the eigenvalue
equations for the quantum integrals $\hat{I}_{i}$

$\Psi(\mathbf{x})=\prod_{i=1}^{N}\psi_{i}(x_{i}),\qquad\tfrac{1}{2}\,\hat{I}_{i}\,\Psi(\mathbf{x})=\nu_{i}\,\Psi(\mathbf{x})\,,$
(17)

in which $\psi(x_{i})\in L^{2}(\mathbb{R},\mathrm{d}x_{i})$, the eigenvalues
$\nu_{i}$ are shown to be PLA ; cq3

$\nu_{i}\equiv\nu_{i}(E,n_{i})=\hbar\Omega\left(n_{i}+\frac{1}{2}\right),\qquad
n_{i}=0,1,2,\dots$ (18)

and the one-particle wave functions $\psi(x_{i})$ are

$\psi_{i}(x_{i})\equiv\psi_{n_{i}}(E,x_{i})=A_{n_{i}}\left(\frac{\beta^{2}}{\pi}\right)^{\frac{1}{4}}\exp\left[-\beta^{2}x_{i}^{2}/2\right]H_{n_{i}}(\beta
x_{i})\,.$ (19)

From these expressions and by taking into account that
${\hat{H}}_{D^{\prime}}=\frac{1}{2}\sum_{i=1}^{N}\hat{I}_{i}$, the discrete
spectrum of the Hamiltonian (15) can be shown to be PLA ; cq3

$\displaystyle\mathcal{E}_{n}=-\hbar^{2}\lambda\left(n+\frac{N}{2}\right)^{2}+\hbar\left(n+\frac{N}{2}\right)\sqrt{\lambda^{2}\hbar^{2}\left(n+\frac{N}{2}\right)^{2}+\omega^{2}}\,,\qquad
n=\sum_{i=1}^{N}n_{i},\quad n=0,1,2\dots$ (20)

Note that the spectrum of the Darboux III oscillator has the very same
degeneracy as the $N$-dimensional quantum isotropic oscillator, but its
eigenvalues are deformed in terms of the curvature parameter $\lambda$, thus
giving rise to a nonlinear spacing among them. This discrete spectrum for the
first six levels of the $N=2$ model is depicted in Fig. 1 for several values
of $\lambda$. As we can observe, by increasing the deformation parameter
$\lambda$, higher eigenvalues become more compressed, albeit the maximal
degeneracy of the full spectrum is preserved.

Figure 1: Schematic representation of the energy levels (20) for the
parameters $\hbar=\omega=1,N=2$ and $\lambda=\\{0,0.02,0.04,0.04,0.06\\}$ with
quantum numbers $n=0,1,\dots,5$. When $\lambda=0$ the energies of the
isotropic harmonic oscillator are recovered.

###  II.3 The 2D Darboux III oscillator coupled to a constant magnetic field

As it is well-known Landau ; Darwin ; Ventriglia , the isotropic oscillator on
the flat 2D Euclidean space under the action of a constant perpendicular
magnetic field in the $z$ direction is also an exactly solvable quantum model.
In fact, if we consider the symmetric gauge

$\mathbf{A}=\left(-\frac{B\,y}{2},\frac{B\,x}{2}\right),$ (21)

the classical Hamiltonian reads

${\mathscr{H}}=\frac{1}{2}\left[\left(p_{x}-\frac{eA_{x}}{c}\right)^{2}+\left(p_{y}-\frac{eA_{y}}{c}\right)^{2}\right]+\frac{1}{2}\omega^{2}(x^{2}+y^{2}),$
(22)

which can be written as

${\mathscr{H}}=\frac{1}{2}\left[(p_{x}^{2}+p_{y}^{2})+\omega_{c}^{2}(x^{2}+y^{2})-2\omega_{c}L_{z}\right]+\frac{1}{2}\omega^{2}(x^{2}+y^{2})\,,$
(23)

where $\omega_{c}=\frac{eB}{2c}$ is the Larmor frequency and
$L_{z}=xp_{y}-yp_{x}$ is the angular momentum in the $z$ direction. Therefore,
by introducing a modulation frequency given by
$\tilde{\omega}^{2}=\omega^{2}_{c}+\omega^{2}$, the Hamiltonian reads

${\mathscr{H}}=\frac{1}{2}\left[(p_{x}^{2}+p_{y}^{2})+\tilde{\omega}^{2}(x^{2}+y^{2})-2\omega_{c}L_{z}\right].$
(24)

The quantum counterpart of this Hamiltonian can be written as

$\hat{\mathscr{H}}=\hat{\mathscr{H}}_{os}+\hat{\mathscr{H}}_{L}=\frac{1}{2}\left[(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})+\tilde{\omega}^{2}(\hat{x}^{2}+\hat{y}^{2})]\right]-\omega_{c}\hat{L}_{z}\,,$
(25)

and since both terms commute
$[\hat{\mathscr{H}}_{os},\hat{\mathscr{H}}_{L}]=0$, the problem can be solved
in terms of simultaneous eigenfunctions of both operators, where the first one
is just the 2D isotropic oscillator with frequency $\tilde{\omega}$. Such
eigenfunctions $\psi_{n,m}$ depend on two quantum numbers which can take
values $n=0,1,2,\dots$ and $m=0.\pm 1,\pm 2,\dots$, and their eigenvalues
$E_{n,m}$ are given by Landau ; Darwin ; Ventriglia

$E_{n,m}=\hbar\,\tilde{\omega}\,(2n+|m|+1)-\hbar\,\omega_{c}\,m.$ (26)

In the following we will show that if we consider that the initial oscillator
has a position-dependent mass of the type
$m(\mathbf{x})=1+\lambda\mathbf{x}^{2}=1+\lambda({x}^{2}+{y}^{2})$, a new
exactly solvable quantum model arises, which can be called he 2D Darboux III
oscillator on a constant magnetic field.

In the presence of an external uniform magnetic field ${B}$, this system would
be defined as

${\mathscr{H}}_{D^{\prime}}=\frac{1}{2(1+\lambda\mathbf{x}^{2})}\left[\left(p_{x}-\frac{eA_{x}}{c}\right)^{2}+\left(p_{y}-\frac{eA_{y}}{c}\right)^{2}+\frac{\omega^{2}}{2}(x^{2}+y^{2})\right],$
(27)

where $e$ is the charge of the particle, and $\mathbf{A}$ is the magnetic
vector potential in the symmetric gauge. When substituting Eq. (21) into Eq.
(27), the Hamiltonian can be written as

${\mathscr{H}}_{D^{\prime}}=\frac{1}{2(1+\lambda\mathbf{x}^{2})}\left[(p_{x}^{2}+p_{y}^{2})+\omega_{c}^{2}(x^{2}+y^{2})-2\omega_{c}L_{z}+\frac{\omega^{2}}{2}(x^{2}+y^{2})\right].$
(28)

Therefore, with the same modulation frequency as above
$\tilde{\omega}^{2}=\omega^{2}_{c}+\omega^{2}$, the Hamiltonian reads

${\mathscr{H}}_{D^{\prime}}=\frac{1}{2(1+\lambda\mathbf{x}^{2})}\left[(p_{x}^{2}+p_{y}^{2})+\tilde{\omega}^{2}(x^{2}+y^{2})-2\omega_{c}L_{z}\right].$
(29)

In the following we show that the quantum version
$\hat{\mathscr{H}}_{D^{\prime}}$ of this model can be also exactly solved by
following the same steps described in Section B for the Darboux oscillator. In
particular, we will have that $\hat{\mathscr{H}}_{D^{\prime}}$, is an
integrable $\lambda$-deformation of the quantum 2D harmonic oscillator
Hamiltonian under the presence of a constant magnetic field.

If we use polar coordinates (see Apendix A), then the Laplacian operator takes
the form

$\nabla^{2}=\partial_{x}^{2}+\partial_{y}^{2}=\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}L_{z}^{2},$
(30)

and under the usual quantization prescription we have
$L_{z}=-i\hbar\partial_{\theta}$. Therefore, the quantum Hamiltonian reads

$\hat{\mathscr{H}}_{D^{\prime}}=\frac{1}{2(1+\lambda
r^{2})}\left[-\hbar^{2}\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\theta}^{2}\right)+2i\hbar\omega_{c}\partial_{\theta}+\tilde{\omega}^{2}r^{2}\right]\,.$
(31)

The wavefunction in polar coordinates $\psi(r,\theta)$ may be factorized as

$\psi(r,\theta)=\frac{1}{\sqrt{r}}u(r)e^{im\theta},$ (32)

with $m\in\mathbb{Z}$, then the eigenvalue equation for the radial function
$u(r)$ arising from Eq. (31) becomes

$\left[-\frac{\hbar^{2}}{2}\partial_{r}^{2}+\frac{\hbar^{2}}{2r^{2}}\left(m^{2}-\frac{1}{4}\right)+\frac{1}{2}\tilde{\Omega}^{2}r^{2}\right]u(r)=\mathscr{E}_{nm}u(r),$
(33)

where
$\tilde{\Omega}=\sqrt{\tilde{\omega}^{2}-2\,{\mathscr{E}}_{nm}\,\lambda}$, and
we will call $\tilde{\mathscr{E}}_{nm}=\mathscr{E}_{nm}+\hbar m\omega_{c}$.
Thus, the eigenspectrum in terms of different quantum numbers $(n,m)$ is
obtained by solving the equation

$\tilde{\mathscr{E}}_{nm}=\hbar\tilde{\Omega}(2n+|m|+1),$ (34)

which leads to the explicit expression for the eigenvalues:

$\mathscr{E}_{nm}=-\hbar
m\omega_{c}-\hbar^{2}\lambda(2n+|m|+1)^{2}+\hbar\sqrt{\left[\hbar\lambda(2n+|m|+1)^{2}+m\omega_{c}\right]^{2}+\tilde{\omega}^{2}(2n+|m|+1)^{2}-m^{2}\omega_{c}^{2}}\,,$
(35)

where the quantum numbers are again $n=0,1,2,\dots$ and $m=0.\pm 1,\pm
2,\dots$.

When the magnetic field vanishes, so that when $\omega_{c}\to 0$, this
expression leads to the eigenvalues obtained in (20) for the Darboux III
Hamiltonian $\hat{H}_{D^{\prime}}$ in two-dimensions. On the other hand, in
the limit $\lambda\to 0$ of constant mass we recover the eigenvalues (26) of
the 2D oscillator within a constant magnetic field. Therefore, this model can
be interpreted as a two-parametric $(\omega_{c},\lambda)$-deformation of the
2D isotropic oscillator. The solutions of Eq. (31) can be written in terms of
the normalized eigenfunctions, namely

$\psi_{n,m}(r,\theta)=\left[\frac{\Gamma(n+|m|)}{\Gamma(n+|m|+1)\pi}\right]^{1/2}\tilde{\beta}^{\tfrac{1}{2}+|m|}r^{|m|}e^{-\frac{\tilde{\beta}^{2}}{2}r^{2}}L_{n}^{|m|}\left(\tilde{\beta}^{2}r^{2}\right)e^{im\theta},\qquad\tilde{\beta}=\frac{\tilde{\Omega}}{\hbar},$
(36)

where $L_{n}^{|m|}(u)$ are generalised Laguerre polynomials.

The lowest energy levels of these exactly solvable quantum systems are plotted
in Figs. 2 and 3. The first one presents the $\lambda=0$ case, i.e., the
Landau levels for a quantum oscillator under in a uniform magnetic field. The
second one shows the influence of the $\lambda$-deformation on such Landau
levels for the cases with both null and non-vanishing magnetic field. Note
that when $B=0$ there is a change of energies for the Landau levels, but they
do not split in terms of $m$ since they correspond to the expression (20) for
the Darboux II oscillator where $n\rightarrow 2n+|m|$ and $N\rightarrow 1$,
and thus the level structure shown in Fig. 1 is reproduced. On the other hand,
when $B\neq 0$ the combined effect of the magnetic field and the position-
dependent mass ruled by $\lambda$ can be appreciated.

Figure 2: The Landau levels are shown in terms of their quantum numbers
$(n,m)$ for the usual case (null magnetic field) and for $B=0.1,0.2,0.5$
(where some of the splittings induced by $m$ arise) by following (26), with
$\hbar=\omega=1$. Note that here $\lambda=0$. Figure 3: Schematic
representation of the Landau levels (35) for the $\lambda-$deformed Darboux
oscillator with quantum numbers $(n,m)$ for the $\lambda-$deformed Darboux
oscillator, and also in conjunction with different magnetic fields of
$B=0.1,0.2,0.5$. Again, we take $\hbar=\omega=1$.

##  III The Dunkl oscillator

###  III.1 The one-dimensional Dunkl oscillator

We recall that the Dunkl oscillator is another exactly solvable quantum system
constructed in terms of Dunkl operators. In the Dunkl quantum mechanics the
momentum operator is expressed in terms of the Dunkl derivative instead of the
ordinary derivative. These operators satisfy the following commutation rules
ito1 ; defin

$\hat{P}_{x}=\frac{\hbar}{i}D_{x}^{\mu_{x}},\qquad\hat{x}=x,\qquad[\hat{x},\hat{P}_{x}]=i\hbar(1+2\mu_{x}R_{x}),$
(37)

where the Dunkl derivative is defined as

$D_{x}^{\mu_{x}}=\partial_{x}+\frac{\mu_{x}}{x}(1-R_{x}),$ (38)

where $\mu_{x}$ is real parameter such that $\mu_{x}>-1/2$, and $R_{x}$
denotes the reflection operator with respect to the plane $x=0$ which acts in
the form $R_{x}f(x)=f(-x)$. The Hamiltonian for the one-dimensional Dunkl
oscillator is defined by

$\hat{\mathcal{H}}_{D}=-\frac{\hbar^{2}}{2}(D_{x}^{\mu_{x}})^{2}+\frac{1}{2}\omega^{2}x^{2}.$
(39)

Since $[\mathcal{H}_{D},R_{x}]=0$, the eigenfunctions $f(x)$ may be chosen to
have a well-defined parity $R_{x}f(x)=e_{x}f(x)$ with $e_{x}=\pm 1$. We can
also say that $q_{x}=0,1$ for $e_{x}=\pm 1$ provided that $q_{x}=(1-e_{x})/2$.
Hence, the eigenfunctions of the Dunkl oscillator are given by

$f_{n_{x}}^{e_{x}}(x)=e^{-\frac{\omega^{2}x^{2}}{2\hbar^{2}}}H_{n_{x}}^{\mu_{x}}(\tfrac{\omega}{\hbar}x),\qquad
n_{x}=0,1,2,\dots$ (40)

where $H_{n_{x}}^{\mu_{x}}(\tfrac{\omega}{\hbar}x)$ are the generalized
Hermite polynomials gener

$H_{n_{x}}^{\mu_{x}}(\tfrac{\omega}{\hbar}x)=(-1)^{n}\sqrt{\frac{\omega
n!}{\hbar\Gamma\left[n+\mu_{x}+q_{x}+\tfrac{1}{2}\right]}}x^{q_{x}}L_{n}^{\left(\mu_{x}+q_{x}-\tfrac{1}{2}\right)}(\tfrac{\omega^{2}}{\hbar^{2}}x^{2}),$
(41)

and the energies of the model are given by the eigenvalues

$\kappa_{n_{x}}^{e_{x}}\equiv\kappa(E^{\prime},n_{x},e_{x})=\hbar\omega\left(n_{x}+\mu_{x}+\frac{1}{2}\right).$
(42)

We recall that the eigenfunctions of the one-dimensional Dunkl oscillator are
normalized and orthogonal on the weighted $L^{2}$ space endowed with the
scalar product iner

$\langle g|f\rangle=\int\overline{g(x)}f(x)|x|^{2\mu_{x}}\mathrm{d}x.$ (43)

###  III.2 The ND Dunkl oscillator

The Dunkl oscillator can be generalized to $N$ dimensions wigner ; ito1 ; ito2
; ito3 ; ndimen . In doing so, the Dunkl momentum operator $\hat{\mathbf{P}}$
associated with the reflection group $\mathbb{Z}^{N}_{2}$ is defined by

$\hat{\mathbf{P}}=\left(\hat{P}_{1},\dots,\hat{P}_{N}\right)=-i\hbar\left(D_{1}^{\mu_{x_{1}}},D_{x_{2}}^{\mu_{2}},\dots,D_{x_{N}}^{\mu_{N}}\right).$
(44)

In fact, we can write

$\displaystyle\nabla_{D}=\left(D_{1}^{\mu_{1}},\dots,D_{N}^{\mu_{N}}\right)=\sum_{i=1}^{N}\partial_{x_{i}}+\frac{\mu_{i}}{x_{i}}(\mathbb{I}-\mathbf{R}_{i}),$
(45)

where $\mathbb{I}$ is the identity operator, $\mu_{i}>-1/2$ are real numbers
and the operator $\mathbf{R}_{i}$ is the reflection operator with respect to
the plane $x_{i}=0$, namely

$\mathbf{R}_{i}f(x_{i})=f(-x_{i}).$ (46)

The Dunkl-Laplacian operator for $\mathbb{Z}^{N}_{2}$ is therefore defined by

$\displaystyle\Delta_{D}=\nabla^{2}_{D}=(D_{1}^{\mu_{1}})^{2}+\dots+(D_{N}^{\mu_{N}})^{2}=\sum_{i=1}^{N}\partial_{x_{i}}^{2}+2\frac{\mu_{i}}{{x_{i}}}\partial_{x_{i}}-\frac{\mu_{i}}{{x_{i}}^{2}}(1-R_{i}).$
(47)

This is a second-order differential-difference operator which reduces to the
usual Laplacian
$\nabla=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}+\dots+\partial_{x_{N}}^{2}$,
when $\mu_{1}=\mu_{2}=\dots=\mu_{N}=0$. The Hamiltonian of the corresponding
$N$-dimensional Dunkl oscillator problem is given by

$\hat{H}_{D}=-\frac{\hbar^{2}}{2}\nabla^{2}_{D}+\frac{1}{2}\omega^{2}\mathbf{x}^{2}\,,$
(48)

and since this system is separable, the related eigenfunctions and energy
spectrum are given in the form

$\displaystyle
F_{n_{x_{1}},n_{x_{2}},\dots,n_{x_{N}}}^{e_{1},e_{2},\dots,e_{N}}(\mathbf{x})=\prod_{i=1}^{N}f_{n_{x_{i}}}^{e_{i}}(x_{i}),$
(49)
$\displaystyle\bar{\mathcal{E}}_{n_{x_{1}},n_{x_{2}},\dots,n_{x_{N}}}^{e_{1},e_{2},\dots,e_{N}}=\sum_{i=1}^{N}\nu_{n_{x_{i}}}^{e_{i}}=\hbar\omega\left(\mathbf{n_{x}}+\mathbf{\mu_{x}}+\frac{N}{2}\right),$
(50)

in which $\mathbf{n_{x}}=\sum_{i=1}^{N}n_{x_{i}}$ and
$\mathbf{\mu_{x}}=\sum_{i=1}^{N}\mu_{i}$.

###  III.3 The 2D Dunkl oscillator coupled to a constant magnetic field

In this section we will describe the 2D Dunkl oscillator in a constant
magnetic field aligned along the z-axis, which will give rise to the Dunkl
generalization of the Landau levels. The model Hamiltonian will be defined as

$\displaystyle\hat{\mathscr{H}}_{D}=-\frac{\hbar^{2}}{2}\nabla_{D}^{2}+i\hbar\omega_{c}\left(xD_{y}^{\mu_{y}}-yD_{x}^{\mu_{x}}\right)+\frac{\tilde{\omega}^{2}}{2}(x^{2}+y^{2}),$
(51)

where $\nabla_{D}^{2}=(D_{x}^{\mu_{x}})^{2}+(D_{y}^{\mu_{y}})^{2}$, and the
Dunkl derivatives $D_{x_{i}}$ are substituted instead of the standard partial
derivatives $\partial_{x_{i}}$. By making use of polar coordinates, the Dunkl-
Laplacian operator can be written as

$\nabla_{D}^{2}=\partial_{r}^{2}+\frac{1}{r}\left(1+2\mu_{x}+2\mu_{y}\right)\partial_{r}-\frac{1}{r^{2}}\left(\mathcal{J}^{2}-2\mu_{x}\mu_{y}(1-R_{x}R_{y})\right),$
(52)

where the angular momentum operator is $\mathcal{J}\equiv
i\hbar(xD_{y}^{\mu_{y}}-yD_{x}^{\mu_{x}})$ so that
$\mathcal{J}^{2}=H_{\theta}+2\mu_{x}\mu_{y}(1-R_{x}R_{y})$, and $H_{\theta}$
is given in Eq. (89) of Appendix A. By substituting Eq. (52) and the
definition of the operator $\mathcal{J}$ into the Hamiltonian (51), the
following operator arises in polar coordinates:

$\hat{\mathscr{H}}_{D}=-\frac{\hbar^{2}}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\left(1+2\mu_{x}+2\mu_{y}\right)\frac{\partial}{\partial
r}\right)+\frac{\hbar^{2}}{2r^{2}}\left(\mathcal{J}^{2}-2\mu_{x}\mu_{y}(1-R_{x}R_{y})\right)+\frac{\tilde{\omega}^{2}}{2}r^{2}+\hbar\omega_{c}\mathcal{J}.$
(53)

In the following we present the solutions of the two-dimensional Dunkl
oscillator in the presence of the magnetic field. Our results are based on the
fact that the operators $R_{x},R_{y}$ commute with the one-dimensional
Hamiltonians $\hat{\mathscr{H}}_{x},\hat{\mathscr{H}}_{y}$, respectively.
Since reflection symmetry is understood as an element of a finite group
$\mathbb{Z}_{2}$, having two irreducible representations for modes with even
and odd parity, then the full Hilbert space of the system is splitted into
even and odd sectors. Therefore, the resulting eigenfunctions for the two-
dimensional Dunkl oscillator can be chosen in such a way that they have a
definite parity, as it is the case by considering the following two
possibilities:

####  III.3.1 Case $R_{x}=R_{y}$

If both sectors are of the same type, $e_{x}=e_{y}=1$ or $e_{x}=e_{y}=-1$,
this leads to $\epsilon\equiv e_{x}e_{y}=1$, where $e_{x}$ and $e_{y}$ are the
eigenvalues of $R_{x}$ and $R_{y}$, respectively. By using the results of
Appendix A, $\mathcal{J}F_{+}=\sigma_{+}F_{+}$ where
$F_{+}=X_{m^{\prime}}^{++}(\theta)\pm iX_{m^{\prime}}^{--}(\theta)$ and
$\sigma_{+}=\pm 2\sqrt{m^{\prime}(m^{\prime}+\mu_{x}+\mu_{y})},\
m^{\prime}\in\mathbb{N}$, where $X_{m^{\prime}}^{++}$ and
$X_{m^{\prime}}^{--}$ are explicitly given in Eqs. (94) and (95).

For $e_{x}=e_{y}=1$, we have $\psi^{++}(r,\theta)=R^{++}(r)F_{+}(\theta)$. In
order to find the radial solutions, we set $R^{++}(r)$ in the following
eigenvalue equation

$\frac{1}{2}\left[-\hbar^{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}(1+2\mu_{x}+2\mu_{y})\frac{\partial}{\partial
r}+\frac{\sigma_{+}^{2}}{r^{2}}\right)+\tilde{\omega}^{2}r^{2}\right]R^{++}(r)=\tilde{\mathscr{E}}^{+}R^{++}(r),$
(54)

where $\tilde{\mathscr{E}}^{+}=\mathscr{E}^{+}-\hbar\omega_{c}\sigma_{+}$.
Therefore, the radial solutions for the coupled Hamiltonian (53) are

$R^{++}_{n}(r)=r^{-\mu_{x}-\mu_{y}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}e^{-\frac{\omega^{2}}{2\hbar^{2}}r^{2}}L_{n}^{\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}(\tfrac{\omega^{2}}{\hbar^{2}}r^{2}),$
(55)

and the energy spectrum is given by the explicit expression

$\mathscr{E}^{+}_{nm^{\prime}}=\hbar\tilde{\omega}\left(2n+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}+1\right)+\hbar\omega_{c}\sigma_{+}.$
(56)

In the same manner, for $e_{x}=e_{y}=-1$, the wave function is
$\psi^{--}(r,\theta)=R^{--}(r)F_{+}(\theta)$ with the same
$\mathcal{J}F_{+}=\sigma_{+}F_{+}$. Consequently, the radial eigensolutions of
(53) are again given by

$R^{--}_{n}(r)=r^{-\mu_{x}-\mu_{y}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}e^{-\frac{\beta^{2}}{2\hbar^{2}}r^{2}}L_{n}^{\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}(\tfrac{\omega^{2}}{\hbar^{2}}r^{2}),$
(57)

and also the eigenvalues are the same as in (56).

####  III.3.2 Case $R_{x}=-R_{y}$

Here, we shall consider $\epsilon\equiv e_{x}e_{y}=-1$, and there are two
possibilities: either $e_{x}=-1,e_{y}=1$ or $e_{x}=1,=e_{y}=-1$. As the
previous case, from Appendix A, we have that
$\mathcal{J}F_{-}=\sigma_{-}F_{-}$, where
$F_{-}=X_{m^{\prime}}^{+-}(\theta)\mp iX_{m^{\prime}}^{-+}(\theta)$ in which
$X_{m^{\prime}}^{+-}(\theta)$ and $X_{m^{\prime}}^{-+}(\theta)$ are given by
Eqs. (97) and (98) of the Appendix A, so that we have $\sigma_{-}=\pm
2\sqrt{(m^{\prime}+\mu_{x})(m^{\prime}+\mu_{y})},\
m^{\prime}\in\\{\tfrac{1}{2},\tfrac{3}{2},\dots\\}$.

For the case $e_{x}=-1$ and $e_{y}=1$, we find that the Hamiltonian (53) leads
to the equation

$\frac{1}{2}\left[-\hbar^{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}(1+2\mu_{x}+2\mu_{y})\frac{\partial}{\partial
r}+\frac{(\sigma_{-}^{2}-4\mu_{x}\mu_{y})}{r^{2}}\right)+\tilde{\omega}^{2}r^{2}\right]R^{+-}(r)=\tilde{\mathscr{E}}^{-}R^{+-}(r)$
(58)

where we have defined
$\tilde{\mathscr{E}}^{-}=\mathscr{E}^{-}-\hbar\omega_{c}\sigma_{-}$. As in the
previous case, Eq. (58) has admissible solutions
$\psi^{+-}(r,\theta)=R^{+-}(r)F_{-}(\theta)$ where the radial solutions
$R^{+-}(r)$ are obtained by solving Eq. (58), and read

$R^{+-}_{n}(r)=r^{-\mu_{x}-\mu_{y}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}}e^{-\frac{\omega^{2}}{2\hbar^{2}}r^{2}}L_{n}^{\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}}(\tfrac{\omega^{2}}{\hbar^{2}}r^{2})$
(59)

together with their eigenvalues

$\mathscr{E}^{-}_{nm^{\prime}}=\hbar\tilde{\omega}\left(2n+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}+1\right)+\hbar\omega_{c}\sigma_{-}.$
(60)

Similarly, for case For $e_{x}=1$ and $e_{y}=-1$ from Eq. (58), we obtain
$R^{+-}(r)=R^{-+}(r)$ with the same eigenvalues
$\mathscr{E}^{-}_{nm^{\prime}}$. The first few energy levels of the Dunkl
oscillator systems are drawn in Fig. 4. It provides the Landau levels for the
Dunkl oscillator under a uniform magnetic field, for which the deformed
parameter is set as $\lambda=0$. These levels are the cases with both null and
non-vanishing magnetic fields. Moreover, under the parity,
$e_{x}e_{y}=\epsilon$, we present the numerical results to the energies given
in (56) and (60) in Fig. 4, which separates the spectrum of the system into
two-level forms, even and odd. For both, it is seen the states with the same
energy $N\rightarrow 2(n+|m^{\prime}|)$, which these degeneracies can be
broken to the different values in presence of the magnetic field.

Figure 4: Schematic representation of the first few Landau levels (56) and
(60) with quantum numbers $(n,m^{\prime})$ for the null magnetic field (the
usual Dunkl oscillator) and for $B=0.1,0.2,0.5$. Note that here $\lambda=0$,
$\hbar=\omega=1$ and the parameter values $\mu_{x}=\mu_{y}=0.02$. The left
figure of the levels are those with the even sector $e_{x}e_{y}=+1$ and the
right figure of the levels are those with the odd sector $e_{x}e_{y}=-1$.

##  IV The Dunkl-Darboux III oscillator

###  IV.1 The one-dimensional model

At this point it seems natural to introduce a model called Dunkl-Darboux III
oscillator $(\hat{\mathcal{H}}_{DD^{\prime}})$ through the Hamiltonian

$\hat{\mathcal{H}}_{DD^{\prime}}=\frac{1}{(1+\lambda
x^{2})}\left[-\frac{\hbar^{2}}{2}(D_{x}^{\mu_{x}})+\frac{\omega^{2}}{2}x^{2}\right]=\frac{1}{(1+\lambda
x^{2})}\hat{\mathcal{H}}_{D}\,,$ (61)

which we will show to be a new exactly solvable quantum model. This model
describes a nonlinear oscillator with a momentum operator given in (37), and
can be interpreted as a new $\lambda-$deformation of the one-dimensional Dunkl
oscillator. We recall that this deformation can be interpreted either as the
motion on a space with nonconstant curvature or as the introduction of a
position-dependent mass. Note that when $\mu_{x}=0$, the Hamiltonian (61)
corresponds to the Darboux III quantum oscillator (6), and when
$\lambda\rightarrow 0$, the Hamiltonian $\hat{\mathcal{H}}_{DD^{\prime}}$
turns out to be the Dunkl oscillator given in (39). Indeed, when both
$\mu_{x}$ and $\lambda$ vanish, the Hamiltonian (61) is just the isotropic
harmonic oscillator.

As a first relevant fact, it turns out that $\hat{\mathcal{H}}_{DD^{\prime}}$
is formally self-adjoint on the $L^{2}$ Hilbert space defined by the scalar
product

$\langle\varphi|\chi\rangle=\int\overline{\varphi(x)}\chi(x)(1+\lambda
x^{2})|x|^{2\mu_{x}}\mathrm{d}x.$ (62)

In the following we obtain the spectrum and eigenfunctions of the Dunkl-
Darboux III quantum oscillator by solving the Schrödinger equation coming from
the Hamiltonian (61), namely

$\displaystyle\hat{\mathcal{H}}_{DD^{\prime}}\,\varphi(x)$
$\displaystyle=\frac{1}{(1+\lambda
x^{2})}\,\hat{\mathcal{H}}_{D}\,\varphi(x)=E^{\prime\prime}\varphi(x),$ (63)
$\displaystyle\hat{\mathcal{H}}_{D}\,\varphi(x)$ $\displaystyle=(1+\lambda
x^{2})\,E^{\prime\prime}\,\varphi(x).$ (64)

Similarly to what we had for the Darboux III oscillator (8), we are led to
consider the following equation

$\frac{1}{2}\left[-\hbar^{2}(D_{x}^{\mu_{x}})^{2}+(\omega^{2}-2E^{\prime\prime}\lambda)x^{2}\right]\varphi(x)=E^{\prime\prime}\varphi(x),$
(65)

and we can define the new energy-dependent frequency

$\Omega^{\prime}(E^{\prime\prime})=\sqrt{\omega^{2}-2E^{\prime\prime}\lambda}\quad\mathrm{where}\quad\omega^{2}>2E^{\prime\prime}\lambda.$
(66)

Hence, eigenfunctions of $\hat{\mathcal{H}}_{DD^{\prime}}$ can be expressed as

$f_{n_{x}}^{e_{x}}(x)=e^{-\frac{\beta^{\prime
2}x^{2}}{2}}H_{n_{x}}^{\mu_{x}}(\beta^{\prime}x),\qquad n_{x}=0,1,2,\dots$
(67)

in which

$H_{n_{x}}^{\mu_{x}}(\beta^{\prime}x)=(-1)^{n}\sqrt{\frac{\beta^{\prime
2}n!}{\Gamma\left[n+\mu_{x}+q_{x}+\tfrac{1}{2}\right]}}x^{q_{x}}L_{n}^{\left(\mu_{x}+q_{x}-\tfrac{1}{2}\right)}(\beta^{\prime
2}x^{2}),\qquad\beta^{\prime}=\sqrt{\frac{\Omega^{\prime}}{\hbar}}.$ (68)

And the energy spectrum is explicitly obtained by solving the equation

$\xi_{n_{x}}^{e_{x}}\equiv\xi(E^{\prime\prime},n_{x},e_{x})=\hbar\Omega^{\prime}\left(n_{x}+\mu_{x}+\frac{1}{2}\right).$
(69)

in terms of $E^{\prime\prime}$, which gives an expression identical to (11)
with $n\rightarrow n_{x}+\mu_{x}$.

###  IV.2 The ND Dunkl-Darboux III oscillator

This section is devoted to the N-dimensional generalization of the Dunkl-
Darboux III oscillator Hamiltonian, which is defined as

$\hat{\mathcal{H}}_{DD^{\prime}}=\frac{1}{(1+\lambda\mathbf{x}^{2})}\left[-\frac{\hbar^{2}}{2}\Delta_{D}+\frac{\omega^{2}}{2}\mathbf{x}^{2}\right]=\frac{1}{(1+\lambda\mathbf{x}^{2})}\,\hat{\mathcal{H}_{D}}\,.$
(70)

By following the same approach as in the previous subsection, the eigenvalue
problem can be written as

$\varPhi_{n_{x_{1}},n_{x_{2}},\dots,n_{x_{N}},\lambda}^{e_{1},e_{2},\dots,e_{N}}(\mathbf{x})=\prod_{i=1}^{N}\varphi_{n_{x_{i}},\lambda}^{e_{i}}(x_{i}),\qquad(\hat{\mathcal{H}}_{DD^{\prime}})_{i}\varPhi=\xi_{n_{x_{i}}}^{e_{i}}\varPhi\,.$
(71)

By taking into account the wave functions of Eq. (67), eigenvalues turn out to
be

$\displaystyle\xi_{n_{x_{i}},\lambda}^{e_{i}}\equiv\xi_{n_{x_{i}},\lambda}^{e_{i}}(E^{\prime\prime},n_{x_{i}})=\hbar\Omega^{\prime}\left(\mathbf{n_{x_{i}}}+\mu_{i}+\frac{N}{2}\right),$
(72)

where once again
$\Omega^{\prime}(E^{\prime\prime})=\sqrt{\omega^{2}-2E^{\prime\prime}\lambda}$.
In this way we obtain the following explicit expression for the energies of
the ND Dunkl-Darboux III oscillator
$\sum_{i=1}^{N}\xi_{n_{x_{i}}}^{e_{i}}=\bar{\mathcal{E}}_{n_{x_{1}},n_{x_{2}},\dots,n_{x_{N}},\lambda}^{e_{1},e_{2},\dots,e_{N}}$
in the form

$\bar{\mathcal{E}}_{n_{x_{1}},n_{x_{2}},\dots,n_{x_{N}},\lambda}^{e_{1},e_{2},\dots,e_{N}}=-\hbar^{2}\lambda\left(\mathbf{n_{x}}+\mathbf{\mu_{x}}+\frac{N}{2}\right)^{2}+\hbar\left(\mathbf{n_{x}}+\mathbf{\mu_{x}}+\frac{N}{2}\right)\sqrt{\lambda^{2}\hbar^{2}\left(\mathbf{n_{x}}+\mathbf{\mu_{x}}+\frac{N}{2}\right)^{2}+\omega^{2}}.$
(73)

###  IV.3 The Dunkl-Darboux III oscillator coupled to a constant magnetic
field

In two dimensions, the Hamiltonian representing the Dunkl-Darboux III
oscillator under a constant magnetic field aligned along the $z$-axis would be
given by

$\hat{\mathscr{H}}_{DD^{\prime}}=\frac{1}{2(1+\lambda\mathbf{x}^{2})}\left[-\hbar^{2}\nabla_{D}^{2}+2\hbar\omega_{c}\mathcal{J}+\tilde{\omega}^{2}(x^{2}+y^{2})\right].$
(74)

By making use of Eq. (52), the radial Hamiltonian takes the form

$\hat{\mathscr{H}}_{DD^{\prime}}=\frac{1}{(1+\lambda
r^{2})}\left[-\frac{\hbar^{2}}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\left(1+2\mu_{x}+2\mu_{y}\right)\frac{\partial}{\partial
r}\right)+\frac{\hbar^{2}}{2r^{2}}\left(\mathcal{J}^{2}-2\mu_{x}\mu_{y}(1-R_{x}R_{y})\right)+\hbar\omega_{c}\mathcal{J}+\frac{\tilde{\omega}^{2}}{2}r^{2}\right].$
(75)

It can be easily checked that
$[R_{x},\hat{\mathscr{H}}_{DD^{\prime}}]=[R_{y},\hat{\mathscr{H}}_{DD^{\prime}}]=0$,
and $[R_{x}R_{y},\hat{\mathscr{H}}_{DD^{\prime}}]=0$, and the operator
$R_{x}R_{y}$ also commutes with the operator $\mathcal{J}$. Therefore we can
make use of the results obtained in Sec. III.3. It what follows, we divide the
radial solutions of Eq. (75) separately in two cases, namely $e_{x}e_{y}=\pm
1$.

####  IV.3.1 Case $R_{x}=R_{y}$

In this case, $e_{x}e_{y}=1$, by substituting the eigenvalue of the operator
$\mathcal{J}$ equal to $\sigma_{+}$ in Eq. (75), we can solve the radial
Dunkl-Darboux III equation, which is

$\frac{1}{2}\left[-\hbar^{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}(1+2\mu_{x}+2\mu_{y})\frac{\partial}{\partial
r}+\frac{\sigma_{+}^{2}}{r^{2}}\right)+\tilde{\Omega}^{2}r^{2}\right]\mathscr{R}^{++}(r)=\tilde{\mathcal{E}}^{+}\mathscr{R}^{++}(r),$
(76)

where $\tilde{\Omega}=\sqrt{\tilde{\omega}^{2}-2\lambda\mathcal{E}^{+}}$ and
$\tilde{\mathcal{E}}^{+}=\mathcal{E}^{+}-\hbar\omega_{L}\sigma_{+}$. Hence,
the energy spectrum of the Dunkl-Darboux III Hamiltonian given in (75) can be
obtained by solving the equation

$\tilde{\mathcal{E}}^{+}_{\tilde{k}}=\hbar\tilde{\Omega}\left(2\tilde{k}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}+1\right).$
(77)

We immediately show that the energy spectrum is

$\displaystyle\mathcal{E}^{+}_{\tilde{k}}$
$\displaystyle=\hbar\omega_{c}\sigma_{+}-\hbar^{2}\lambda\left(2\tilde{k}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}+1\right)^{2}$
(78)
$\displaystyle+\hbar\sqrt{\left(\hbar\lambda\left(2\tilde{k}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}+1\right)^{2}-\omega_{c}\sigma_{+}\right)^{2}+\tilde{\omega}^{2}\left(2\tilde{k}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}+1\right)^{2}-\omega_{c}^{2}\sigma_{+}^{2}},$

with the eigenfunctions of the Dunkl-Darboux III oscillator given as

$\mathscr{R}^{++}_{\tilde{k}}(r)=r^{-\mu_{x}-\mu_{y}+\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}\exp\left[-\frac{\beta^{2}}{2}r^{2}\right]L_{\tilde{k}}^{\sqrt{(\mu_{x}+\mu_{y})^{2}+\sigma_{+}^{2}}}(\beta^{2}r^{2}),\qquad\beta=\sqrt{\frac{\tilde{\Omega}}{\hbar}}\,.$
(79)

In order to obtain the explicit form
$\varphi^{--}_{\tilde{k}^{\prime}}(r,\theta)=\mathscr{R}_{\tilde{k}^{\prime}}^{--}(r)F_{+}(\theta)$
and its energy spectrum with two odd indexes, we should use Eqs. (77) and (79)
with the different quantum number $\tilde{k}^{\prime}$.

####  IV.3.2 Case $R_{x}=-R_{y}$

Finally, we consider the second possibility given by $\epsilon=e_{x}e_{y}=-1$.
Thus, from equation (75) we find that the radial part of the eigenfunctions
$\psi^{+-}_{\tilde{\ell}}(r,\theta)=\mathscr{R}^{-+}_{\tilde{\ell}}(r)F_{-}(\theta)$
are given by the equation

$\frac{1}{2}\left[-\hbar^{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}(1+2\mu_{x}+2\mu_{y})\frac{\partial}{\partial
r}+\frac{(\sigma_{-}^{2}-4\mu_{x}\mu_{y})}{r^{2}}\right)+\tilde{\Omega}^{2}r^{2}\right]\mathscr{R}^{+-}_{\tilde{\ell}}(r)=\tilde{\mathcal{E}}^{-}_{\tilde{\ell}}\mathscr{R}^{+-}_{\tilde{\ell}}(r),$
(80)

where now
$\tilde{\Omega}_{\tilde{\ell}}=\sqrt{\tilde{\omega}^{2}-2\lambda\mathcal{E}}^{-}_{\tilde{\ell}}$
and
$\tilde{\mathcal{E}}^{-}_{\tilde{\ell}}=\mathcal{E}^{-}_{\tilde{\ell}}-\hbar\omega_{L}\sigma_{-}$.
Therefore, in this case the energy spectrum
$\tilde{\mathcal{E}}^{-}_{\tilde{\ell}}$ is

$\tilde{\mathcal{E}}^{-}_{\tilde{\ell}}=\hbar\tilde{\Omega}\left(2\tilde{\ell}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}+1\right),$
(81)

Substituting the expression for $\tilde{\Omega}_{\tilde{\ell}}$, the
eigenvalues are found to be

$\displaystyle\mathcal{E}^{-}_{\tilde{\ell}}$
$\displaystyle=\hbar\omega_{c}\sigma_{-}-\hbar^{2}\lambda\left(2\tilde{\ell}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}+1\right)^{2}$
(82)
$\displaystyle+\hbar\sqrt{\left(\hbar\lambda\left(2\tilde{\ell}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}+1\right)^{2}-\omega_{c}\sigma_{-}\right)^{2}+\tilde{\omega}^{2}\left(2\tilde{\ell}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}+1\right)^{2}-\omega_{c}^{2}\sigma_{-}^{2}}.$

The explicit expression of eigenfunctions in terms of generalized Laguerre
polynomials is

$\mathscr{R}^{+-}_{\tilde{\ell}}(r)=r^{-\mu_{x}-\mu_{y}+\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}}\exp\left[-\frac{\beta^{2}}{2}r^{2}\right]L_{\tilde{\ell}}^{\sqrt{(\mu_{x}-\mu_{y})^{2}+\sigma_{-}^{2}}}(\beta^{2}r^{2}).$
(83)

Under the condition $e_{x}=-1$ and $e_{y}=1$, we find that the solutions of
the eigenfunctions
$\psi^{-+}_{\tilde{\ell}^{\prime}}(r,\theta)=\mathscr{R}^{-+}_{\tilde{\ell}^{\prime}}(r)F_{-}(\theta)$
are the same given in Eq. (83) with the eigenvalues of
$\mathcal{E}^{-}_{\tilde{\ell}}$ given in terms of the quantum number
$\tilde{\ell}^{\prime}$. In Fig. 5, we have drawn numerical values for the
first energy in the new model of the Dunkl-Darboux III, which is a combination
of the two previous models. Plots of the Landau levels indicate the influence
of the $\lambda-$deformation parameter for the Dunkl-Darboux III oscillator in
the presence and absence of the magnetic field. It is enough for one of the
parameters $\mu_{i},B_{i}$, and $\lambda$ to go to zero, in this case, we will
gain one of the modes discussed in the previous sections.

Figure 5: Schematic representation of the first few Landau levels (78) and
(82) with quantum numbers $(n,m^{\prime})$ for the $\lambda-$deformed Dunkl-
Darboux oscillator with the different magnetic fields $B=0.1,0.2,0.5$, where
the parameters are set with $\hbar=\omega=1$ and $\mu_{x}=\mu_{y}=0.02$. The
left figure of the levels are those with the even sector $e_{x}e_{y}=+1$ and
the right figure of the levels are those with the odd sector $e_{x}e_{y}=-1$.

##  V Conclusions

Summarizing, in this paper we have introduced a new familly of exactly
solvable quantum models in $N$-dimensions which are constructed as integrable
deformations of the isotropic oscillator Hamiltonian in two different, albeit
compatible, directions. In the first one, a conformally flat space with
nonconstant curvature is introduced as the background space for the model. On
the other, the discrete symmetries provided by Dunkl operators can be also
simultaneously introduced. The deformed symmetries allowed by both approaches
turn out to be also compatible, and provide all the tools needed in order to
solve the Dunkl-Darboux III model explicitly. Finally, in the 2D case a
constant magnetic field can be also included by preserving the exact
solvability of the Dunkl-Darboux III Hamiltonian. Moreover, the role played by
the curvature parameter, the Dunkl eigenvalues and the magnetic field in the
energies and degeneracies of the corresponding Landau levels can be explicitly
shown.

##  Appendix A N-dimensional hyperspherical coordinates and Dunkl derivatives

In analyzing the Darboux systems generalized with Dunkl derivatives on spaces
of nonconstant curvature, the generalization to $N$-dimensional spaces of the
hyperspherical coordinates $r,\theta_{j}$ is needed (see cq2  for details).
They are formed by a radial type coordinate $r=|\mathbf{x}|\in\mathbb{R}^{+}$
and $N-1$ angles $\theta_{j}$ such that $\theta_{k}\in[0,2\pi)$ for $k<N-1$
and $\theta_{N-1}\in[0,\pi)$. The coordinates are defined in correspondence
with the Cartesian coordinates:

$\hat{x}_{j}=\hat{r}\cos\hat{\theta}_{j}\prod_{k=1}^{j-1}\sin\hat{\theta}_{k},\quad
1\leq j<N\qquad\hat{x}_{N}=\hat{r}\prod_{k=1}^{N-1}\sin\hat{\theta}_{k},$ (84)

Hence the metric (1) takes the form

$\mathrm{d}s^{2}=(1+\lambda
r^{2})(\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}),\qquad\mathrm{d}\Omega^{2}=\sum_{j=1}^{N-1}\mathrm{d}\theta_{j}^{2}\prod_{k=1}^{j-1}\sin^{2}\theta_{k}.$
(85)

Now, by considering the introduction of the quantum operators
$p_{r}=-i\partial_{r},p_{\theta_{j}}=-i\partial_{\theta_{j}}$, the
corresponding Dunkl derivatives are defined as

$\displaystyle D_{j}$
$\displaystyle=\prod_{k=1}^{j-1}\sin\hat{\theta}_{k}\cos\hat{\theta}_{j}\partial_{r}+\frac{\cos\hat{\theta}_{j}}{\hat{r}}\sum_{i=1}^{j-1}\frac{\prod_{k=l+1}^{j-1}\sin\hat{\theta}_{k}}{\prod_{m=1}^{l-1}\sin\hat{\theta}_{m}}\cos\hat{\theta}_{l}\partial_{\theta_{l}}-\frac{\sin\hat{\theta}_{j}}{\hat{r}\prod_{k=1}^{j-1}\sin\hat{\theta}_{k}}\partial_{\theta_{j}}+\frac{\mu_{j}}{\hat{r}\cos\hat{\theta}_{j}\prod_{k=1}^{j-1}\sin\hat{\theta}_{k}}(1-R_{j}),$
(86) $\displaystyle D_{N}$
$\displaystyle=\prod_{k=1}^{N-1}\sin\hat{\theta}_{k}\partial_{r}+\frac{1}{\hat{r}}\sum_{l=1}^{N-1}\frac{\prod_{k=l+1}^{N-1}\sin\hat{\theta}_{k}}{\prod_{m=1}^{l-1}\sin\hat{\theta}_{m}}\cos\hat{\theta}_{l}\partial_{\theta_{l}}+\frac{\mu_{N}}{\hat{r}\prod_{k=1}^{N-1}\sin\hat{\theta}_{k}}(1-R_{N}).$
(87)

Then the explicit expression for the N-dimensional generalization of the Dunkl
operators (86) allows to write the Dunkl-Laplacian operator as

$\Delta_{D}=\frac{\partial^{2}}{\partial
r^{2}}+\frac{2\mu_{j}+N-1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}H_{\theta},$ (88)

where the operator $H_{\theta}$ is given in the two-dimensional case by

$\displaystyle
H_{\theta}=-\frac{\partial^{2}}{\partial\theta^{2}}+\left(\mu_{x}\tan-\mu_{y}\cot\theta\right)\frac{\partial}{\partial\theta}+\frac{\mu_{x}(1-R_{x})}{\cos^{2}\theta}+\frac{\mu_{y}(1-R_{y})}{\sin^{2}\theta},$
(89)

so that $H_{\theta}$ is related to the square of the Dunkl operator
$\mathcal{J}$ in the following way

$\mathcal{J}^{2}=H_{\theta}+2\mu_{x}\mu_{y}(1-R_{x}R_{y}),$ (90)

where in polar coordinates the operator $\mathcal{J}$ takes the form

$\mathcal{J}=i\left(\partial_{\theta}+\mu_{y}\cot\theta(1-R_{y})-\mu_{x}\tan\theta(1-R_{x})\right).$
(91)

From the results presented in Refs. dirac2 ; pse ; ito1 , we summarize the
results for the eigenvalues of the angular momentum. In this way, we have the
eigenvalue equation in the following form

$\mathcal{J}F_{\epsilon}=\sigma_{\epsilon}F_{\epsilon},$ (92)

in which $\epsilon\equiv e_{x}e_{y}=\pm 1$, and $e_{x},e_{y}$ are the
eigenvalues of the reflection operators $R_{x}$ and $R_{y}$, respectively.
Since $\epsilon=\pm 1$, eigenvalues and eigenfunctions of $\mathcal{J}$ can be
obtained for $R_{x}=\epsilon R_{y}$. When $R_{x}=R_{y}$, that is $\epsilon=1$,
the solutions of Eq. (92) of $\mathcal{J}$ are given by

$F_{+}=X^{++}_{m^{\prime}}(\theta)\pm iX^{--}_{m^{\prime}}(\theta)$ (93)

with

$\displaystyle X^{++}_{m^{\prime}}(\theta)$
$\displaystyle=\sqrt{\frac{(2m^{\prime}+\mu_{x}+\mu_{y})\Gamma(m^{\prime}+\mu_{x}+\mu_{y})m^{\prime}!}{2\Gamma(m^{\prime}+\mu_{x}+1/2)\Gamma(m^{\prime}+\mu_{y}+1/2)}}P^{(\mu_{x}-1/2,\mu_{y}-1/2)}_{m^{\prime}}(x),$
(94) $\displaystyle X^{--}_{m^{\prime}}(\theta)$
$\displaystyle=\sqrt{\frac{(2m^{\prime}+\mu_{x}+\mu_{y})\Gamma(m^{\prime}+\mu_{x}+\mu_{y})(m^{\prime}-1)!}{2\Gamma(m^{\prime}+\mu_{x}+1/2)\Gamma(m^{\prime}+\mu_{y}+1/2)}}\sin\theta\cos\theta
P^{(\mu_{x}+1/2,\mu_{y}+1/2)}_{m^{\prime}-1}(x),$ (95)

where $m^{\prime}$ is a non-negative integer, $\sigma_{+}=\pm
2\sqrt{m^{\prime}(m^{\prime}+\mu_{x}+\mu_{y})}$, and
$P^{(\alpha,\beta)}_{m^{\prime}}(x)$ are the Jacobi polynomials with $x=-\cos
2\theta$. While for $R_{x}=-R_{y}(\epsilon=-1)$, the solutions of
$\mathcal{J}$ leads to the expression

$F_{-}=X^{-+}_{m^{\prime}}(\theta)\mp iX^{+-}_{m^{\prime}}(\theta)$ (96)

with

$\displaystyle X^{+-}_{m^{\prime}}(\theta)$
$\displaystyle=\sqrt{\frac{(2m^{\prime}+\mu_{x}+\mu_{y})\Gamma(m^{\prime}+\mu_{x}+\mu_{y}+1/2)(m^{\prime}-1/2)!}{2\Gamma(m^{\prime}+\mu_{x})\Gamma(m^{\prime}+\mu_{y}+1)}}\sin\theta
P^{(\mu_{x}-1/2,\mu_{y}+1/2)}_{m^{\prime}-1/2}(x),$ (97) $\displaystyle
X^{-+}_{m^{\prime}}(\theta)$
$\displaystyle=\sqrt{\frac{(2m^{\prime}+\mu_{x}+\mu_{y})\Gamma(m^{\prime}+\mu_{x}+\mu_{y}+1/2)(m^{\prime}-1/2)!}{2\Gamma(m^{\prime}+\mu_{x}+1)\Gamma(m^{\prime}+\mu_{y})}}\cos\theta
P^{(\mu_{x}+1/2,\mu_{y}-1/2)}_{m^{\prime}-1/2}(x).$ (98)

where $m^{\prime}$ is a positive half-integer, and $\sigma_{-}=\pm
2\sqrt{(m^{\prime}+\mu_{x})(m^{\prime}+\mu_{y})}$.

## Acknowledgements

This work was partially supported by Agencia Estatal de Investigación (Spain)
under grant PID2019-106802GB-I00/AEI/10.13039/501100011033 (A.B.) and
20220355-SIP-IPN, Mexico (SH.D.). A. N. would like to thank the members of the
Mathematical Physics Research Group of the University of Burgos for their kind
assistance and hospitality.

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