Supersymmetric Quesne-Dunkl Quantum Mechanics on Radial Lines

Abstract

Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.

1. Introduction

Quantum deformations have long been a central theme in theoretical physics, offering deep understanding into various aspects of quantum mechanics. A prominent topic within this domain is the reflection operator R, whose origins can be traced back to the foundational work of Wigner [1] and Yang [2]. In recent years, the Dunkl oscillator has gained significant attention as it incorporates Dunkl operators [3,4] into the harmonic oscillator model, thus enriching its underlying algebraic framework [5,6,7,8].

A notable extension of this framework is the $C_{\lambda }$-extended oscillator algebra, which is associated with the cyclic group $C_{\lambda }$ of order $\lambda$. This was first introduced by C. Quesne and N. Vansteenkiste [6]. This algebraic structure allows for a clear formulation of creation and annihilation operators through difference-differential operators and facilitates the study of coherent states. A similar approach employing difference-differential operators was further developed in [8]. More recently, a generalized version of the Dunkl oscillator, built upon Milovanović’s generalized Hermite polynomials along radial rays, was introduced in [9,10].

On a related note, Supersymmetric Quantum Mechanics (SSQM) has emerged as a powerful framework for uncovering symmetries between bosonic and fermionic systems [11,12]. When integrated with the concept of shape invariance [13], SSQM becomes a versatile tool for constructing exactly solvable quantum mechanical models. Despite the success of these methods, the pursuit of new approaches for generating shape-invariant potentials continues to be an area of active research. In particular, Refs. [14,15] proposed a formulation of supersymmetric quantum mechanics for one-dimensional systems that employs Yang-Dunkl operators [2]. A key aspect of this approach is that both the supersymmetric Hamiltonian and the supercharges involve reflection operators, with wave functions expressed in terms of Hermite orthogonal polynomials.

A central focus of the study of generalized Hermite polynomials on radial rays is understanding their link to supersymmetric quantum mechanics. In this work, we introduce a novel approach centered on a complex reflection operator. Specifically, we define a new type of Dunkl operator based on this generalized reflection operator. By analyzing the resulting system, we seek to gain deeper insights into the Dunkl oscillator and its connection to Hermite orthogonal polynomials on radial lines.

2. The Generalized Hermite Polynomials on the Radial Lines

The concept of generalized Hermite polynomials on radial lines was initially introduced by Milovanović in [9,10]. These polynomials are expressed in terms of generalized Hermite polynomials as follows [16]:

$${\mathcal{H}}_n^{(r,\nu )}\left(x\right)=x^s{H}_m^{\left({\nu }_s\right)}\left(x^r\right),$$

(1)

where

$$n=mr+s,{\nu }_s=\nu +{\displaystyle \frac{2s+1}{2r}},s=0,\dots ,r-1,$$

(2)

and $H_m^{\left(\nu \right)}\left(x\right)$ represents the generalized Hermite polynomial [17] (formula 2.43, p. 156). These generalized Hermite polynomials $H_m^{\left(\nu \right)}\left(x\right)$ can be expressed in terms of the Laguerre polynomials as follows:

$$\left\{\begin{array}{c}{H}_{2m}^{\left(\nu \right)}\left(x\right)={(-1)}^m{2}^{2m}m!L_m^{(\nu -{\displaystyle \frac{1}{2}})}\left(x^2\right),\hfill \\ H_{2m+1}^{\left(\nu \right)}\left(x\right)={(-1)}^m{2}^{2m+1}m!x{L}_m^{(\nu +{\displaystyle \frac{1}{2}})}\left(x^2\right),\hfill \end{array}\right.$$

(3)

where $L_m^{\nu }\left(x\right)$ is the Laguerre polynomial [18], defined as:

$$L_m^{\nu }\left(x\right)={{\displaystyle \frac{{(\nu +1)}_m}{m!}}}_1{F}_1\left({\displaystyle \genfrac{}{}{0pt}{}{-m}{\nu +1}};x\right)={\displaystyle \frac{{(\nu +1)}_m}{m!}}\sum _{k=0}^m{\displaystyle \frac{{(-m)}_k}{{(\nu +1)}_k}}{\displaystyle \frac{x^k}{k!}}.$$

They satisfy the following orthogonality relations [16]:

$$\sum _{j=0}^{r-1}{\int }_{\mathbb{R}}{\mathcal{H}}_n^{(r,\nu )}\left({\omega }_r^{j}x\right)\overline{{\mathcal{H}}_{n^{\prime }}^{(r,\nu )}\left({\omega }_r^{j}x\right)}{\left|x\right|}^{2r\nu }{e}^{-x^{2r}}dx={\zeta }_n{\delta }_{n{n}^{\prime }},{\omega }_r=e^{{\displaystyle \frac{2i\pi }{r}}},\nu >-{\displaystyle \frac{1}{2r}},$$

where

$${\zeta }_n=\left\{\begin{array}{cc}{2}^{4m}m!\Gamma \left(m+\nu +{\displaystyle \frac{1+2s}{2r}}\right),\hfill & \mathrm{if}n=2mr+s,\hfill \\ 2^{4m+2}m!\Gamma \left(m+1+\nu +{\displaystyle \frac{1+2s}{2r}}\right),\hfill & \mathrm{if}n=(2m+1)r+s,\hfill \end{array}\right.$$

(4)

with $s=0,1,\dots ,r-1$.

In this study, it is crucial to highlight the use of a different set of orthogonal polynomials from those used by Milovanović. While we employ the generalized Hermite polynomials defined in (3), Milovanović’s method is rooted in the use of generalized Laguerre polynomials, as discussed in [9,10].

We define the deformed number ${\left[n\right]}_{\nu }$ for each $n=0,1,\dots$ as follows:

$${\left[n\right]}_{\nu }=⌊{\displaystyle \frac{n}{r}}⌋+{\vartheta }_n,$$

where

$${\vartheta }_n=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & \mathrm{if}n=2mr+s,s=0,\dots ,r-1,\hfill \\ {\displaystyle 2{\nu }_s,}\hfill & \mathrm{if}n=2mr+s,s=r,\dots ,2r-1.\hfill \end{array}\right.$$

Here, $⌊·⌋$ denotes the floor function, which rounds down to the nearest integer.

It is evident that

$${\left[n\right]}_{\nu }={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\zeta }_n}{{\zeta }_{n-r}}}.$$

(5)

With our notation, the three-term recurrence relations are expressed as follows [16]:

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{n+r}^{(r,\nu )}\left(x\right)=2x^r{\mathcal{H}}_n^{(r,\nu )}\left(x\right)-2{\left[n\right]}_{\nu }{\mathcal{H}}_{n-r}^{(r,\nu )}\left(x\right),n\ge r,\hfill \\ \hfill & {\mathcal{H}}_n^{(r,\nu )}\left(x\right)=2^{⌊{\displaystyle \frac{n}{r}}⌋}{x}^n,n=0,1,\dots ,r-1.\hfill \end{array}$$

The wave function for the harmonic Generalized Dunkl oscillator was explicitly presented in [16] as:

$${\psi }_n^{(r,\nu )}\left(x\right)={\zeta }_n^{-1/2}{e}^{-{\displaystyle \frac{x^{2r}}{2}}}{\mathcal{H}}_n^{(r,\nu )}\left(x\right),$$

where the normalization constant ${\zeta }_n$ is defined in (4).

These wave functions satisfy the following orthogonality relations:

$$\sum _{j=0}^{r-1}{\int }_{\mathbb{R}}{\psi }_n^{(r,\nu )}\left({\omega }_r^{j}x\right)\overline{{\psi }_{n^{\prime }}^{(r,\nu )}\left({\omega }_r^{j}x\right)}{\left|x\right|}^{2r\nu }dx={\delta }_{n{n}^{\prime }}.$$

We introduce the complex reflection operator S of order $2r$$(r\in \mathbb{N})$, acting on a function f as:

$$\left(Sf\right)\left(x\right):=f\left({\epsilon }_{r}x\right),{\epsilon }_r=e^{{\displaystyle \frac{i\pi }{r}}}.$$

(6)

This operator generates the cyclic group $C_r=\{1,S,S^2,\dots ,S^{2r-1}\}$, and the corresponding orthogonal projections ${\pi }_0,{\pi }_1,\dots ,{\pi }_{2r-1}$ are given by:

$${\pi }_j={\displaystyle \frac{1}{2r}}\sum _{k=0}^{2r-1}{\epsilon }_r^{-kj}{S}^k,j=0,\dots ,2r-1.$$

These operators satisfy:

$$\sum _{j=0}^{2r-1}{\pi }_j=1,{\pi }_i{\pi }_j={\delta }_{ij}{\pi }_i.$$

We adopt the conventions ${\pi }_{j^{\prime }}={\pi }_j$ if $j^{\prime }-j\equiv 0mod2r$.

We define the differential-difference operator:

$$Y_{\nu }={\displaystyle \frac{1}{r{x}^{r-1}}}{\displaystyle \frac{d}{dx}}+{\displaystyle \frac{1}{x^r}}\sum _{k=1}^{2r-1}{\gamma }_k{\pi }_k,$$

(7)

where:

$${\gamma }_k=\left\{\begin{array}{cc}-{\displaystyle \frac{k}{r}},\hfill & \mathrm{if}k=1,\dots ,r-1,\hfill \\ 2\nu +{\displaystyle \frac{1+k-r}{r}},\hfill & \mathrm{if}k=r,\dots ,2r-1.\hfill \end{array}\right.$$

The raising and lowering operators for the generalized Hermite polynomials on radial lines are given by [16]:

$$Y_{\nu }{\mathcal{H}}_n^{(r,\nu )}\left(x\right)=2{\left[n\right]}_{\nu }{\mathcal{H}}_{n-r}^{(r,\nu )}\left(x\right),$$

(8)

and

$$\begin{array}{c}\hfill \left(Y_{\nu }+2x^r\right){\mathcal{H}}_n^{(r,\nu )}\left(x\right)={\mathcal{H}}_{n+r}^{(r,\nu )}\left(x\right).\end{array}$$

(9)

The generalized Hermite polynomials also satisfy the following differential-difference equation:

$$\begin{array}{c}\hfill Y_{\nu }(Y_{\nu }+2x^r){\mathcal{H}}_n^{(r,\nu )}\left(x\right)=2{\left[n\right]}_{\nu }{\mathcal{H}}_n^{(r,\nu )}\left(x\right).\end{array}$$

From (8) and (9), we deduce

$$\begin{array}{cc}\hfill \left((Y_{\nu }+x^r)\right){\psi }_n^{(r,\nu )}\left(x\right)& =\sqrt{2{\left[n\right]}_{\nu }}{\psi }_{n-r}^{(r,\nu )}\left(x\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill (-Y_{\nu }+x^r){\psi }_n^{(r,\nu )}\left(x\right)& =\sqrt{2{[n+r]}_{\nu }}{\psi }_{n+r}^{(r,\nu )}\left(x\right).\hfill \end{array}$$

(11)

The annihilation a and creation $a^{†}$ operators were explicitly introduced in [16] as:

$$\begin{array}{c}\hfill a={\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\nu }+x^r\right),a^{†}={\displaystyle \frac{1}{\sqrt{2}}}\left(-Y_{\nu }+x^r\right).\end{array}$$

These operators satisfy the following commutation relations:

$$\begin{array}{cc}\hfill [a,a^{†}]& =1+\sum _{j=0}^{2r-1}{\widehat{\kappa }}_j{\pi }_j,a^{†}{\pi }_j={\pi }_{j+r}{a}^{†}.\hfill \end{array}$$

(12)

where the constants ${\kappa }_s$ are defined as:

$$\widehat{{\kappa }_s}=\left\{\begin{array}{cc}2\nu +{\displaystyle \frac{2s+1}{r}},\hfill & \mathrm{if}s=0,\dots ,r-1,\hfill \\ 2-2\nu -{\displaystyle \frac{2s+1}{r}},\hfill & \mathrm{if}s=r,\dots ,2r-1.\hfill \end{array}\right.$$

Inspired by relation (12), we define the algebra generated by the operators 1, $a^{†}$, $a={\left(a^{†}\right)}^{†}$, $N=N^{†}$, and $S={\left(S^{†}\right)}^{-1}$, which satisfy the following commutation relations:

$$\begin{array}{cc}\hfill [N,a^{†}]& =a^{†},[N,{\pi }_j]=0,\sum _{j=0}^{2r-1}{\pi }_j=1,\hfill \\ \hfill [a,a^{†}]& =1+\sum _{j=0}^{2r-1}{\widehat{\kappa }}_j{\pi }_j,a^{†}{\pi }_j={\pi }_{j+r}{a}^{†},\hfill \end{array}$$

(13)

where ${\widehat{\kappa }}_j$, for $j=0,1,2,\dots ,2r-1$, are explicitly given by (2). As is conventional, the operators N, $a^{†}$, and a are referred to as the number, creation, and annihilation operators, respectively. This algebra, denoted by $Q\left(2r\right)$, will be referred to as the Quesne-Dunkl oscillator algebra. It is important to note that the Quesne-Dunkl oscillator algebra $Q\left(2r\right)$ represents a slight modification of the Extended Heisenberg algebra [6]. The key distinction between these two algebras lies in the final relation of (13), where the Quesne-Dunkl oscillator satisfies $a^{†}{\pi }_j={\pi }_{j+r}{a}^{†}$, while in the Extended Heisenberg algebra, the corresponding relation is $a^{†}{\pi }_j={\pi }_{j+1}{a}^{†}$. We can introduce the complex reflections $S^k,k=0,\dots ,2r-1,$ by:

$$S^k=\sum _{j=0}^{2r-1}{e}^{\pi ikj/r}{\pi }_j.$$

(14)

and conversely primitive idempotents ${\pi }_0,{\pi }_1,\dots ,{\pi }_{2r-1}$, may be expressed in terms of the ${\pi }_j$’s as

$${\pi }_k={\displaystyle \frac{1}{2r}}\sum _{j=0}^{2r-1}{e}^{-2\pi ikj/2r}{S}^j,$$

By utilizing the complex reflection operator S defined in (17), the extended oscillator algebra can be reformulated in terms of the operators 1, $a^{†}$, $a={\left(a^{†}\right)}^{†}$, $N=N^{†}$, and $S={\left(S^{†}\right)}^{-1}$, which satisfy the following relations:

$$\begin{array}{cc}\hfill [N,a^{†}]& =a^{†},[N,S]=0,S^{2r}=1,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill [a,a^{†}]& =1+\sum _{j=1}^{2r-1}{\kappa }_j{S}^j,a^{†}S=-S{a}^{†},\hfill \end{array}$$

(16)

along with their Hermitian conjugates. Here, the parameters ${\kappa }_j$ for $j=1,2,\dots ,2r-1$ are complex and satisfy ${\kappa }_j^{*}={\kappa }_{2r-j}$, and are related to ${\widehat{\kappa }}_j$ by

$${\kappa }_j={\displaystyle \frac{1}{2r}}\sum _{l=0}^{2r-1}{e}^{{\displaystyle \frac{\pi ilj}{r}}}{\widehat{\kappa }}_l.$$

Similar to the structure of the Extended Heisenberg algebra [6], the $Q\left(2r\right)$ algebra can be understood as a generalized deformed oscillator algebra (GDOA), where

$$G\left(N\right)=1+\sum _{j=0}^{2r-1}{\widehat{\kappa }}_j{\pi }_j.$$

In the context of any GDOA, one can define a structure function $F\left(N\right)$, which satisfies the difference equation

$$F(N+1)-F\left(N\right)=G\left(N\right),$$

with the initial condition $F\left(0\right)=0$.

As demonstrated in [6], GDOAs generally allow for various unitary irreducible representations based on the nature of the spectrum of N. However, our focus here is on the bosonic Fock-space representation, where the Fock space $\mathcal{F}$ is spanned by the eigenstates $|n\rangle$ of the number operator N, with eigenvalues $n=0,1,2,\dots$, and where $|0\rangle$ is the vacuum state, such that $a|0\rangle =0$. The projection operator ${\pi }_l$ projects onto the l-th component ${\mathcal{F}}_l\equiv \left\{\right|2rk+l\rangle |k=0,1,2,\dots \}$ of the ${\mathbb{Z}}_{2r}$-graded Fock space $\mathcal{F}={⨁}_{l=0}^{2r-1}{\mathcal{F}}_l$.

From the relations (10) and (2), the creation and annihilation operators act on $|n\rangle$ as:

$$a^{†}|n\rangle =\sqrt{{[n+r]}_{\nu }}|n+r\rangle ,a|n\rangle =\sqrt{{[n-r]}_{\nu }}|n-r\rangle .$$

For $n=mr+s$, the eigenvectors $|n\rangle$ can be written as

$$|n\rangle ={\mathcal{N}}_n^{-1/2}{\left(a^{†}\right)}^m|0\rangle ,$$

where ${\mathcal{N}}_n$ is a normalization factor, given by

$${\mathcal{N}}_n={\left({[mr+s]}_{\nu }!!\right)}^{-1/2},$$

with

$${[mr+s]}_{\nu }!!=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}m=0,\hfill \\ {\left[r+s\right]}_{\nu }\dots {\left[mr+s\right]}_{\nu },\hfill & \mathrm{for}m\in \mathbb{N}.\hfill \end{array}\right.$$

It is obvious that such a Fock-space representation exists if and only if

$$\nu >-{\displaystyle \frac{3}{2r}}.$$

In the bosonic Fock space representation, we may consider the bosonic oscillator Hamiltonian, defined as usual by

$$H_{r,\nu }={\displaystyle \frac{1}{2}}(a{a}^{†}+a^{†}a).$$

It can be rewritten as

$$H_{r,\nu }=a^{†}a+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\sum _{s=0}^{2r-1}\widehat{{\kappa }_s}{\pi }_s.$$

It thus immediately follows that

$$H_{r,\nu }|n\rangle ={\displaystyle \frac{1}{2}}({[n+r]}_{\nu }+{\left[n\right]}_{\nu })|n\rangle ,n=0,1,2\dots .$$

(17)

3. Supersymmetric Quantum Mechanics on Radial Rays

A Hamiltonian H is considered as supersymmetric if there exist supercharges Q and $Q^{†}$ that satisfy the following superalgebra relations [12]:

$$[Q,H]=0,[Q^{†},H]=0,H=\{Q,Q^{†}\},$$

where $[a,b]:=ab-ba$ represents the commutator, and $\{a,b\}:=ab+ba$ denotes the anticommutator. The supercharges Q and $Q^{†}$ are Hermitian conjugates. When the supercharge Q is self-adjoint, i.e., $Q^{*}=Q^{†}$, it follows that $H=2Q$, based on these relations. As established in [15], this framework is referred to as $N={\displaystyle \frac{1}{2}}$ supersymmetric quantum mechanics. In [15], the authors introduced several realizations of $N={\displaystyle \frac{1}{2}}$ supersymmetric quantum mechanics in one dimension, where the supercharge is represented as a difference-differential operator of Dunkl type:

$$Q={\displaystyle \frac{1}{\sqrt{2}}}\left({\partial }_{x}R+u\left(x\right)R+v\left(x\right)\right),$$

(18)

where $u\left(x\right)$ is an even function, $v\left(x\right)$ is an odd function, and R is the reflection operator, acting on a function $f\left(x\right)$ as:

$$Rf\left(x\right)=f(-x).$$

Our aim in this section is to explore a new model for $N={\displaystyle \frac{1}{2}}$-supersymmetric quantum mechanics. To achieve this, we replace the standard derivative ${\partial }_x$ in the realization of the supercharge given by (18) with the operator $Y_{\nu }$ defined in (7). Thus, a realization of one-dimensional supersymmetric quantum mechanics on radial lines is obtained by taking the supercharge Q to be the following differential-difference operator of Dunkl type:

$$Q={\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\nu }+U\left(x\right)\right)R+{\displaystyle \frac{1}{\sqrt{2}}}V\left(x\right),$$

(19)

where R is the reflection operator defined as:

$$R=\sum _{s=0}^{r-1}({\pi }_s-{\pi }_{r+s}),R^2=1,$$

(20)

and the functions $U\left(x\right)$ and $V\left(x\right)$ satisfy the following relations:

$$RU=UR,RV=-VR.$$

Since $R^{*}=R,$ it is easy to see that

$$\begin{array}{c}\hfill Y_{\nu }{R}_r=-R_r{Y}_{\nu },x^{r}R=-R{x}^r,R^2=1.\end{array}$$

A straightforward computation shows that Q shares that properties

$$Q^{*}=Q.$$

After evaluating $Q^2$, we get the following form for a supersymmetric Hamiltonian:

$$\widehat{H}=Q^2=-{\displaystyle \frac{1}{2}}{Y}_{\nu }^2+{\displaystyle \frac{1}{2}}{U}^2+{\displaystyle \frac{1}{2}}{V}^2+{\displaystyle \frac{1}{2}}[Y_{\nu },U]-{\displaystyle \frac{1}{2}}R[Y_{\nu },V].$$

Consider a simple example derived from (19) by setting

$$U\left(x\right)=0,V\left(x\right)=x^r.$$

This yields the supercharge operator:

$$Q={\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\nu }R+x^r\right).$$

(21)

The spectrum of the Hamiltonian ${\widehat{H}}_{r,\nu }$ is readily obtained by observing that

$$\begin{array}{cc}\hfill {\widehat{H}}_{r,\nu }& =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d^2}{d{x}^2}}+{\displaystyle \frac{1}{2}}{x}^{2r}-{\displaystyle \frac{1}{2}}R-{\displaystyle \frac{1}{2}}R\sum _{s=0}^{r-1}\left(2\nu +{\displaystyle \frac{2s+1}{r}}\right)\left({\pi }_s-{\pi }_{r+s}\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d^2}{d{x}^2}}+{\displaystyle \frac{1}{2}}{x}^{2r}-{\displaystyle \frac{1}{2}}R-{\displaystyle \frac{1}{2}}\sum _{s=0}^{r-1}\left(2\nu +{\displaystyle \frac{2s+1}{r}}\right)\left({\pi }_s+{\pi }_{r+s}\right).\hfill \end{array}$$

(22)

Hence, for $m=0,1,2,\dots$ and $s=0,1,\dots ,r-1$, we have

$$\begin{array}{cc}\hfill {\widehat{H}}_{r,\nu }|2mr+s\rangle & =2n|2mr+s\rangle ,s=0,1,\dots ,r-1,\hfill \\ \hfill {\widehat{H}}_{r,\nu }|(2m+1)r+s\rangle & =(2m+2)\left|\right(2n+2)r+s\rangle .\hfill \end{array}$$

Additionally, observe that the operator Q can be expressed as:

$$Q={\displaystyle \frac{1}{2}}\left\{(1+R)a^{†}+(1-R)a\right\}.$$

It follows that:

$$Q|n\rangle =\left\{\begin{array}{cc}\sqrt{{\left[n\right]}_{\nu }}|n-r\rangle ,\hfill & \mathrm{if}⌊{\displaystyle \frac{n}{r}}⌋\mathrm{is}\mathrm{even},\hfill \\ \\ \sqrt{{[n+r]}_{\nu }}|n+r\rangle ,\hfill & \mathrm{if}⌊{\displaystyle \frac{n}{r}}⌋\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(23)

In view of (23), it is immediately clear that the states

$$|2mr+s,ϵ\rangle ={\displaystyle \frac{1}{2}}\left(\left|\right(2m+1)r+s\rangle +ϵ|(2m+2)r+s\rangle \right),ϵ=\pm 1,$$

(24)

satisfy the following:

$$Q|2mr+s,ϵ\rangle =ϵ\sqrt{2n+2}|2mr+s,ϵ\rangle .$$

Thus, it immediately follows that:

$${\widehat{H}}_{r,\nu }|2mr+s,ϵ\rangle =(2m+2)|2mr+s,ϵ\rangle ,{\widehat{H}}_{r,\nu }|0\rangle =0$$

The spectrum of ${\widehat{H}}_{r,\nu }$ consists solely of even numbers starting from zero. Each energy level is degenerate, with the exception of the ground state, which is non-degenerate. As demonstrated in [14], a minor modification of the supercharge (21) and the Hamiltonian (22) allows for the construction of a quantum system in the phase of broken supersymmetry. This leads to a system with a doubly degenerate, equidistant positive energy spectrum, governed by a linear superalgebra of the form (22).

It is well established that, in the coordinate representation, the wave functions $\langle x|n,\nu \rangle$ are expressed in terms of the generalized Hermite polynomials $H_n^{\nu }\left(x\right)$.

$$\langle x|n\rangle ={\zeta }_n^{-1/2}{e}^{-{\displaystyle \frac{x^{2r}}{2}}}{\mathcal{H}}_n^{(r,\nu )}\left(x\right),$$

where the normalization constant is defined by

$${\zeta }_n=\left\{\begin{array}{cc}{2}^{4m}m!\Gamma \left(m+\nu +{\displaystyle \frac{1+2k}{2r}}\right),\hfill & \mathrm{if}n=2mr+k,\hfill \\ 2^{4m+2}m!\Gamma \left(m+1+\nu +{\displaystyle \frac{1+2k}{2r}}\right),\hfill & \mathrm{if}n=(2m+1)r+k,\hfill \end{array}\right.$$

From (24), we have

$$\langle x|2mr+s,ϵ\rangle ={\varrho }_{m,s}{e}^{-x^{2r}/2}\left\{ϵH_{2m+2}^{\left({\nu }_s\right)}\left(x^r\right)+2\sqrt{m+1}{H}_{2m+1}^{\left({\nu }_s\right)}\left(x^r\right)\right\},$$

where

$${\varrho }_{m,s}={\displaystyle \frac{1}{2{(2^{4m+4}(m+1)!\Gamma (m+1+\nu +\frac{1+2s}{2r})]}^{1/2}}}.$$

It is evident from this example that the operator R, defined in Equation (20), maps degenerate eigenstates into one another:

$$R|n,ϵ\rangle =-|n,-ϵ\rangle .$$

This behavior follows from the specific properties:

$$\{Q,R\}=0,[{\widehat{H}}_{r,\nu },R]=0.$$

As a consequence, R, which is diagonalized simultaneously with ${\widehat{H}}_{r,\nu }$, transforms an eigenstate of Q with eigenvalue $ϵ\sqrt{2m+2}$ into another eigenstate of Q, degenerate in energy, with eigenvalue $-ϵ\sqrt{2m+2}$. This accounts for the two-fold degeneracy observed in the system’s energy levels, except for the ground state.

4. The Case: $\mathit{r}=\mathbf{1}$

In the particular case $r=1$ and $\nu >-{\displaystyle \frac{1}{2}}$, the supercharge Q and Hamiltonian $\widehat{H}$, defined respectively in (21) and (22), take the forms:

$$\begin{array}{cc}\hfill Q& ={\displaystyle \frac{1}{\sqrt{2}}}\left(D_{\nu }R+x\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \widehat{H}& ={\displaystyle \frac{1}{2}}{\displaystyle \frac{d^2}{d{x}^2}}-{\displaystyle \frac{\nu }{x}}{\displaystyle \frac{d}{dx}}+{\displaystyle \frac{\nu }{2x^2}}(1-R)+{\displaystyle \frac{1}{2}}{x}^2-{\displaystyle \frac{1}{2}}R-\nu .\hfill \end{array}$$

(26)

The spectrum of $\widehat{H}$ is then given by:

$$\widehat{H}|n,\nu \rangle =\left(n+{\displaystyle \frac{1}{2}}(1-{(-1)}^n)\right)|n,\nu \rangle .$$

(27)

Before proceeding deeper into the examination of the spectrum of the supersymmetric Dunkl oscillator $\widehat{H}$ and the supercharge Q, let us recall the $\nu$-deformed Heisenberg algebra. It is given by the generators $a,a^{†},R$ and 1, satisfying the commutation relations:

$$\begin{array}{cc}\hfill & [a,a^{†}]=1+2\nu R,R^2=1,\{a,R\}=0,\{a^{†},R\}=0.\hfill \end{array}$$

Here, the step operators are realized as:

$$a={\displaystyle \frac{1}{\sqrt{2}}}\{D_{\nu }+x\},a^{†}={\displaystyle \frac{1}{\sqrt{2}}}\{-D_{\nu }+x\}.$$

(28)

The Hamiltonian then reads:

$$\begin{array}{c}\hfill H={\displaystyle \frac{1}{2}}\{a{a}^{†}+a^{†}a\}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d^2}{d{x}^2}}-{\displaystyle \frac{\nu }{x}}{\displaystyle \frac{d}{dx}}+{\displaystyle \frac{\nu }{2x^2}}(1-R)+{\displaystyle \frac{1}{2}}{x}^2.\end{array}$$

(29)

Considering the number operator N obeying:

$$[N,a^{†}]=a^{†},[N,a]=-a,$$

we have:

$$a^{†}a={\left[N\right]}_{\nu },a{a}^{†}={[N+1]}_{\nu }.$$

Introducing the Fock space as follows:

$$N|n,\nu \rangle =n|n,\nu \rangle ,n=0,1,2,\dots ,$$

the energy levels are given by:

$$E_n={\displaystyle \frac{1}{2}}({\left[n\right]}_{\nu }+{[n+1]}_{\nu }),n=0,1,2,\dots ,$$

where the $\nu$-numbers are defined by:

$${\left[n\right]}_{\nu }=n+\nu (1-{(-1)}^n),n=0,1,2,\dots .$$

The spectrum of $\widehat{H}$ is easily obtained by observing that:

$$\widehat{H}=H-{\displaystyle \frac{1}{2}}R+\nu .$$

(30)

Given that

$$R|n,\nu \rangle ={(-1)}^n|n,\nu \rangle ,n=0,1,2,\dots ,$$

it follows that:

$$\widehat{H}|n,\nu \rangle =(n+{\displaystyle \frac{1}{2}}(1-{(-1)}^n))|n,\nu \rangle .$$

Let

$$|n,\epsilon ,\nu \rangle ={\displaystyle \frac{1}{2}}\left(|2n,\nu \rangle +\epsilon |2n+1,\nu \rangle \right),\epsilon =\pm 1.$$

Then

$$Q|0,\epsilon ,\nu \rangle =0,Q|n,\epsilon ,\nu \rangle =\epsilon \sqrt{2n+2}|n,\epsilon ,\nu \rangle ,n=1,2,\dots .$$

It immediately follows that

$$\widehat{H}|0,\nu \rangle =0,\widehat{H}|n,ϵ,\nu \rangle =2n+2|n,\epsilon ,\nu \rangle ,n=1,2,\dots$$

As is well known, in the coordinate representation, the wave functions $\langle x|n,\nu \rangle$ are given in terms of the generalized Hermite polynomials $H_n^{\left(\nu \right)}\left(x\right)$ by

$$\langle x|n,\nu \rangle ={\gamma }_n^{-1/2}{e}^{-x^2/2}{H}_n^{\left(\nu \right)}\left(x\right),$$

where the normalization constant is defined as where ${\gamma }_n$ is given by:

$${\gamma }_n=2^{2n}\Gamma (\lfloor {\displaystyle \frac{n}{2}}\rfloor +1)\Gamma (\lfloor {\displaystyle \frac{n+1}{2}})\rfloor +\nu +{\displaystyle \frac{1}{2}}),$$

(31)

From (24), we have

$$\langle x|n,ϵ,\nu \rangle ={\displaystyle \frac{e^{-x^2/2}}{2^{2n+3}{[n!\Gamma (n+1+\nu +1/2)]}^{1/2}}}\left\{H_{2n+2}^{\left(\nu \right)}\left(x\right)+ϵ2\sqrt{n+1}{H}_{2n+1}^{\left(\nu \right)}\left(x\right)\right\}.$$

5. Conclusions

In this work, we have explored the Dunkl oscillator model and its integration into the framework of supersymmetric quantum mechanics (SSQM). By introducing a complex reflection operator, we provided a novel approach to examining the connection between Dunkl operators and Hermite polynomials on radial rays. This analysis has deepened our understanding of the algebraic structure underlying the Dunkl oscillator and its relation to SSQM.

The extended $C_{\lambda }$-oscillator algebra, which links the system to cyclic groups of order $\lambda$, has been instrumental in studying the creation and annihilation operators through difference-differential operators. This approach opens up new possibilities for constructing exactly solvable quantum models, offering further insight into the rich algebraic properties of quantum deformation theory.

Future work may focus on applying this method to other orthogonal polynomials and investigating the extension of these results to relativistic systems and curved spacetime, further expanding the potential applications of the Dunkl oscillator in theoretical physics.