Algebraic derivation of the Energy Eigenvalues for the quantum oscillator defined on the Sphere and the Hyperbolic plane

We discuss a quadratic algebraic approach to the classicalCarinena et al. (2004) and the quantumCarinena, Rañada, and Santander (2007); Rañada (2014); Cariñena, Ranada, and Santander (2007a); Carinena, Ranada, and Santander (2007); Quesne (2015) harmonic oscillators on the sphere and the hyperbolic plane leading ultimately to their spectral properties without invoking the Schrodinger equation; the energy eigenvalues were calculated by using classical means in references [Cariñena, Ranada, and Santander, 2007a] and [Carinena, Ranada, and Santander, 2007, Quesne, 2015] wherein special function solutions of a spectral problem appear. These nonlinear classical and quantum models extend to two spatial dimensions a well known classical nonlinear oscillator Mathews and Lakshmanan (1974); Lakshmanan and Rajaseekar (2012) with an exactly solvable quantum analog Schulze-Halberg and Morris (2012); Cariñena, Ranada, and Santander (2004, 2007b). The non-linearity parameter $\lambda$ which enters the definition of both a nonlinear potential and a position dependent mass in the model of references [Mathews and Lakshmanan, 1974] and [Lakshmanan and Rajaseekar, 2012] can be interpreted in the 2D model of Cariñena et al Carinena et al. (2004) as the negative of the constant Gaussian curvature $\kappa$ of the underlying two-dimensional space (i.e. $\lambda=-\kappa$), thereby showing that their model describes a harmonic oscillator on the sphere for negative $\lambda$ and on the hyperbolic plane for positive $\lambda$.

Cariñena et alCarinena et al. (2004) showed that their 2D system is superintegrable and that its Hamilton-Jacobi equation is separable in three different coordinate system in agreement with the fact that superintegrable systems are Hamiltonian systems with more integrals of motion than the degrees of freedom. In two dimensions, the maximum number of functionally independent integrals is three, and that is the case in reference [Carinena et al., 2004] we are considering. In references [Cariñena, Ranada, and Santander, 2007a,Carinena, Ranada, and Santander, 2007] the energy eigenvalues and wave-function for the quantum analog of this classical model were determined by solving the spectral problem in the corresponding three coordinate systems where the Schrodinger equation becomes separable.

In the present work we shed new light on the 2D classical and quantum models of the harmonic oscillator by following a quantum algebraic approach due to DaskaloyannisDaskaloyannis (2001) to calculate energy eigenvalues of the 2D quadratically superintegrable systems. The method of reference [Daskaloyannis, 2001] fixes the energy eigenvalues by assuming that they can be determined by the existence of a finite-dimensional representation of the polynomial algebra of the motion integral operators. The entire calculation proceeds by using the tool of deformed parafermionic oscillator algebra Quesne (1994). The energy eigenvalues can be found by solving the appropriate algebraic equations with two unknown parameters $u$ and $E$ to be determined. The energy eigenvalues were calculated by the authors of references [Cariñena, Ranada, and Santander, 2007a] and [Carinena, Ranada, and Santander, 2007, Quesne, 2015]. In this article we calculate the eigenvalues of energy and find that these eigenvalues of energy are the same as the eigenvalues which were calculated by classical analytical techniques. It is worth noting a preliminary attempt to calculate the energy spectrum using the analogy to the harmonic oscillator on the Euclidean plane in the reference [Bonatsos, Daskaloyannis, and Kokkotas, 1994].

The structure of the paper is as follows. In Section 2, we consider a Hamiltonian system which is more general than the curved space generalization of the classical harmonic oscillator and review the general form of the quadratic Poisson algebra presented in reference [Daskaloyannis, 2001] for the quadratically superintegrable system. In Section 3, we then apply these general considerations to arrive at the special form of the quadratic Poisson algebra for the 2D nonlinear harmonic oscillator. In Section 4, we give the quantum analogue of the quadratic Poisson algebra considered in Section 2, the corresponding quadratic associative algebra. We then express the Casimir operator for this algebra in terms of the Hamiltonian and give the realization of the quadratic algebra in terms of the deformed parafermionic oscillator algebra. The finite dimensional representations of the quadratic algebra are generated by using the technique of the deformed parafermionic algebra. Finally, the problem is reduced to the solution of a system of two algebraic equations. In Section 5, we give the quadratic associative algebra and the energy eigenvalues for the 2D quantum harmonic oscillator determined by solving the appropriate algebraic equations. In Section 6, we give a precise and a detailed discussion of the comparison with the results of the analytical calculations. Our main conclusions are given at the end of that section.

In this section we consider a general superintegrable system in a two-dimensional space(not necessarily Euclidean) with a scalar potential. We will assume that we have a quadratic Hamiltonian and due to superintegrability the system possesses two more integrals of motion A and B that are quadratic functions of canonical momenta too. Since A and B are constants of motion they satisfy the following Poisson bracket relations,

$\displaystyle\{H,A\}=\{H,B\}=0.$

(2.1)

Since H, A, B are quadratic in momenta, we can expect them to generate a quadratic algebra like that in reference [Daskaloyannis, 2001]. We set

	$\displaystyle\{A,B\}$	$\displaystyle=C,$
	$\displaystyle\{A,C\}$	$\displaystyle=\alpha A^{2}+2\gamma AB+\delta A+\epsilon B+\zeta,$
	$\displaystyle\{B,C\}$	$\displaystyle=aA^{2}+\rho B^{2}+2\sigma AB+dA+\eta B+z.$		(2.2)

Since we have $\{C,\{A,B\}\}=0$ the Jacobi identity reduces to

$\displaystyle\{A,\{B,C\}\}=\{B,\{A,C\}\}.$

(2.3)

Thus the Jacobi identity implies $\rho=-\gamma$, $\sigma=-\alpha$, $\eta=-\delta$ and we obtain the quadratic algebra

	$\displaystyle\{A,B\}$	$\displaystyle=C,$
	$\displaystyle\{A,C\}$	$\displaystyle=\alpha A^{2}+2\gamma AB+\delta A+\epsilon B+\zeta,$
	$\displaystyle\{B,C\}$	$\displaystyle=aA^{2}-\gamma B^{2}-2\alpha AB+dA-\delta B+z.$		(2.4)

The coefficients $a$, $\alpha$ and $\gamma$ are constants, but the other ones can be polynomials in the Hamiltonian H. The degrees of these polynomials are dictated by the fact that $H,A$ and $B$ are second order polynomials in the momenta. Hence, C can be a third order polynomial. We have that $\delta$, $\epsilon$ and $d$ are each equal to at most a linear function of H whereas $\zeta$ and $z$ are each equal to at most a quadratic function of H. A Casimir operator $K$ of a polynomial algebra is defined as a Poisson operator commuting with all the elements of the algebra. For the algebra (2.4) this means that

$\displaystyle\{K,A\}=\{K,B\}=\{K,C\}=0,$

(2.5)

and this implies

$\displaystyle K=C^{2}-2\alpha A^{2}B-2\gamma AB^{2}-2\delta AB-\epsilon B^{2}-2\zeta B+\frac{2}{3}aA^{3}+dA^{2}+2zA.$

(2.6)

Thus K is a polynomial of degree 6 in the momenta. Since the Hamiltonian H also satisfies relations (2.5) we can expect K to be a polynomial in H and we write

$\displaystyle K=k_{0}+k_{1}H+k_{2}H^{2}+k_{3}H^{3},$

(2.7)

where $k_{0}$, $k_{1}$, $k_{2}$ and $k_{3}$ are constants. Note that (2.7) together with (2.6) represents a polynomial relation between the integrals $H$, $A$, $B$ and $C$ in agreement with the fact that the integral of motion $C$ is not independent from the integrals $H$, $A$ and $B$. Therefore the integrals of motion of a quadratically superintegrable two dimensional system close to form the quadratic Poisson algebra (2.4), corresponding to a Casimir equal to at most a cubic function of the Hamiltonian (2.7).

Now let us consider the superintegrable Hamiltonian Carinena et al. (2004) for the curved space generalization of the harmonic oscillator on the 2D sphere and the hyperbolic plane

$\displaystyle H=\frac{1}{2}\left[p_{x}^{2}+p_{y}^{2}-\kappa(xp_{x}+yp_{y})^{2}\right]+\frac{\omega^{2}}{2}\left(\frac{x^{2}+y^{2}}{1-\kappa(x^{2}+y^{2})}\right).$

(3.1)

The Hamiltonian can also be written as

$\displaystyle H=H_{1}+H_{2}+\kappa H_{3},$

(3.2)

where $H_{1}$, $H_{2}$ and $H_{3}$ are the three independent integrals of motion that are given by

	$\displaystyle H_{1}$	$\displaystyle=\frac{1}{2}\left(\left[1-\kappa(x^{2}+y^{2})\right]p_{x}^{2}+\omega^{2}\frac{x^{2}}{1-\kappa(x^{2}+y^{2})}\right),$		(3.3)
	$\displaystyle H_{2}$	$\displaystyle=\frac{1}{2}\left(\left[1-\kappa(x^{2}+y^{2})\right]p_{y}^{2}+\omega^{2}\frac{y^{2}}{1-\kappa(x^{2}+y^{2})}\right),$		(3.4)
	$\displaystyle H_{3}$	$\displaystyle=\frac{1}{2}(xp_{y}-yp_{x})^{2}.$		(3.5)

Taking A as $2H_{1}$ and B as $2H_{2}$ we can identify the classical coefficients which characterize the corresponding algebra (2.4). They are given by

	$\displaystyle\alpha$	$\displaystyle=-8,$
	$\displaystyle\gamma$	$\displaystyle=-8\kappa,$
	$\displaystyle\delta$	$\displaystyle=16H,$
	$\displaystyle\epsilon$	$\displaystyle=-16\omega^{2},$
	$\displaystyle\zeta$	$\displaystyle=a=d=z=0.$		(3.6)

The value of the Casimir calculated from (2.6) is

$\displaystyle K=0.$

(3.7)

It is essential to study the quadratic Poisson algebra and its Casimir as we will observe in the next section that it correspond to the lowest order terms in $\hslash$ of the quadratic associative algebra and its Casimir operator in the corresponding quantum system.

It is worth remarking that the above algebra is between $A$, $B$ and $C$ is the direct consequence of the underlying Poisson algebra between integrals $C_{i}$ defined as follows

	$\displaystyle C_{1}$	$\displaystyle=H,$		(3.8)
	$\displaystyle C_{2}$	$\displaystyle=H_{1}=\frac{1}{2}\left([1-\kappa(x^{2}+y^{2})]p_{x}^{2}+\omega^{2}\frac{x^{2}}{1-\kappa(x^{2}+y^{2})}\right),$		(3.9)
	$\displaystyle C3$	$\displaystyle=H_{2}=\frac{1}{2}\left([1-\kappa(x^{2}+y^{2})]p_{y}^{2}+\omega^{2}\frac{y^{2}}{1-\kappa(x^{2}+y^{2})}\right),$		(3.10)
	$\displaystyle C_{4}$	$\displaystyle=(xp_{y}-yp_{x}),$		(3.11)
	$\displaystyle C_{5}$	$\displaystyle=\left([1-\kappa(x^{2}+y^{2})]p_{x}p_{y}+\omega^{2}\frac{xy}{1-\kappa(x^{2}+y^{2})}\right).$		(3.12)

So, we have the following two constraint functionals

	$\displaystyle F_{1}$	$\displaystyle=C_{1}-\left(C_{2}+C_{3}+\frac{\kappa}{2}C_{4}^{2}\right),$		(3.13)
	$\displaystyle F_{2}$	$\displaystyle=2C_{2}C_{3}-\frac{\alpha^{2}}{2}C_{4}^{2}-\frac{1}{2}C_{5}^{2},$		(3.14)

such that they vanish on the space of solutions. Following TegmanTeĞMen (2006), the non-vanishing Poisson brackets between the $C_{i}$ can be written in terms of the constrained functionals

	$\displaystyle\{C_{2},C_{4}\}$	$\displaystyle=-\frac{\partial(F_{1},F_{2})}{\partial(C_{3},C_{5})}=-C_{5},$		(3.15)
	$\displaystyle\{C_{2},C_{3}\}$	$\displaystyle=\frac{\partial(F_{1},F_{2})}{\partial(C_{4},C_{5})}=\kappa C_{4}C_{5},$		(3.16)
	$\displaystyle\{C_{2},C_{5}\}$	$\displaystyle=\frac{\partial(F_{1},F_{2})}{\partial(C_{3},C_{4})}=\alpha^{2}C_{4}+2\kappa C_{2}C_{4},$		(3.17)
	$\displaystyle\{C_{3},C_{4}\}$	$\displaystyle=\frac{\partial(F_{1},F_{2})}{\partial(C_{2},C_{5})}=C_{5},$		(3.18)
	$\displaystyle\{C_{3},C_{5}\}$	$\displaystyle=-\frac{\partial(F_{1},F_{2})}{\partial(C_{2},C_{4})}=-\alpha^{2}C_{4}-2\kappa C_{3}C_{4},$		(3.19)
	$\displaystyle\{C_{4},C_{5}\}$	$\displaystyle=\frac{\partial(F_{1},F_{2})}{\partial(C_{2},C_{3})}=(2C_{3}-2C_{2}).$		(3.20)

In this section, we review the quantum version of the quadratic Poisson algebra which is the quadratic associative algebra of operators for the quantum 2D quadratically superintegrable systems. The integrals A and B are now independent motion integral operators. The quadratic associative algebra for the generators {A, B, C} generated by motion integral operators of the 2D quadratically superintegrable systems satisfy the following commutation relationsDaskaloyannis (2001)

	$\displaystyle[A,B]$	$\displaystyle=C,$		(4.1)
	$\displaystyle[A,C]$	$\displaystyle=\alpha A^{2}+\gamma\{A,B\}+\delta A+\epsilon B+\zeta,$		(4.2)
	$\displaystyle[B,C]$	$\displaystyle=aA^{2}-\gamma B^{2}-\alpha\{A,B\}+dA-\delta B+z.$		(4.3)

The Casimir operator for this algebra is given by

	$\displaystyle K$	$\displaystyle=C^{2}-\alpha\{A^{2},B\}-\gamma\{A,B^{2}\}$
		$\displaystyle+(\alpha\gamma-\delta)\{A,B\}+(\gamma^{2}-\epsilon)B^{2}+(\gamma\delta-2\zeta)B$
		$\displaystyle+\frac{2}{3}aA^{3}+(d+\frac{a\gamma}{3}+\alpha^{2})A^{2}+(\frac{a\epsilon}{3}+\alpha\delta+2z)A.$		(4.4)

The quadratic algebra Q(3) (4.1), (4.2) and (4.3) is realized in terms of the deformed oscillator algebraDaskaloyannis (2001) $\{\mathcal{N},b^{\dagger},b\}$ satisfying the following equations:

	$\displaystyle[\mathcal{N},b^{\dagger}]$	$\displaystyle=b^{\dagger},$
	$\displaystyle[\mathcal{N},b]$	$\displaystyle=-b,$
	$\displaystyle b^{\dagger}b$	$\displaystyle=-\Phi(\mathcal{N}),$
	$\displaystyle bb^{\dagger}$	$\displaystyle=-\Phi(\mathcal{N}+1),$		(4.5)

where $\mathcal{N}$ is the number operator and $\Phi(x)$ is a well behaved real function known as the structure function satisfying the following conditions

$\displaystyle\Phi(0)=0,\qquad\text{and}\qquad\Phi(x)>0,\qquad\text{for}\qquad x>0.$

(4.6)

A Fock-space type description is possible when we impose the existence of an integer p such that $\Phi(p+1)=0$.The deformed oscillator algebra in this instance is a parafermionic oscillator algebra. The realization of this quadratic algebra Q(3) is of the form

$\displaystyle\text{A}=A(\mathcal{N}),\qquad\text{B}=b(\mathcal{N})+b^{\dagger}\rho(\mathcal{N})+\rho(\mathcal{N})b,$

(4.7)

where the functions $A(x)$, $b(x)$ and $\rho(x)$ are to be determined. By defining

$\displaystyle\Delta A(\mathcal{N})=A(\mathcal{N}+1)-A(\mathcal{N}),$

(4.8)

the generator C is realized as

$\displaystyle C=[A,B]=b^{\dagger}\Delta A(\mathcal{N})\rho(\mathcal{N})-\rho(\mathcal{N})\Delta A(\mathcal{N})b.$

(4.9)

Now by using (4.2), (4.7) we have the following two equations

	$\displaystyle(\Delta A(\mathcal{N}))^{2}=\gamma(A(\mathcal{N}+1)+A(\mathcal{N}))+\epsilon,$		(4.10)
	$\displaystyle\alpha A(\mathcal{N})^{2}+2\gamma A(\mathcal{N})b(\mathcal{N})+\delta A(\mathcal{N})+\epsilon b(\mathcal{N})+\zeta=0.$		(4.11)

The function $A(\mathcal{N})$ is determined from (4.10) which depends on the value of the parameter $\gamma$ while the $b(\mathcal{N})$ is uniquely obtained from the (4.11) provided that atmost one of the parameters $\gamma$ or $\epsilon$ is not zero.

The value of $A(\mathcal{N})$ and $b(\mathcal{N})$ from equation (4.10) and (4.11) for $\gamma\neq 0$ is

	$\displaystyle A(\mathcal{N})$	$\displaystyle=\frac{\gamma}{2}((\mathcal{N}+u)^{2}-\frac{1}{4}-\frac{\epsilon}{\gamma^{2}}),$		(4.12)
	$\displaystyle b(\mathcal{N})$	$\displaystyle=-\frac{\alpha((\mathcal{N}+u)^{2}-\frac{1}{4})}{4}+\frac{\alpha\epsilon-\delta\gamma}{2\gamma^{2}}-\frac{\alpha\epsilon^{2}-2\epsilon\delta\gamma+4\gamma^{2}\zeta}{4\gamma^{4}}\frac{1}{((\mathcal{N}+u)^{2}-\frac{1}{4})},$		(4.13)

where the parameter u is to be determined. For the case when $\gamma=0$ and $\epsilon\neq 0$ we refer the reader to reference [Daskaloyannis, 2001] for the corresponding formulas.

Using (4.3), (4.4) and the derived value of $A(\mathcal{N})$ and $b(\mathcal{N})$ we arrive at the following two equations

	$$\displaystyle 2\Phi(\mathcal{N}+1)(\Delta A(\mathcal{N})+\frac{\gamma}{2})\rho(\mathcal{N})-2\Phi(\mathcal{N})(\Delta A(\mathcal{N}-1)-\frac{\gamma}{2})\rho(\mathcal{N}-1)$$
	$$\displaystyle=aA^{2}(\mathcal{N})-\gamma b^{2}(\mathcal{N})-2\alpha A(\mathcal{N})b(\mathcal{N})+dA(\mathcal{N})-\delta b(\mathcal{N})+z,$$		(4.14)
and,
	$$\displaystyle\begin{split}K={}&\Phi(\mathcal{N}+1)(\gamma^{2}-\epsilon-2\gamma A(\mathcal{N})\\ &-\Delta A^{2}(\mathcal{N}))\rho(\mathcal{N})+\Phi(\mathcal{N})(\gamma^{2}-\epsilon-2\gamma A(\mathcal{N})\\ &-\Delta A^{2}(\mathcal{N}-1))\rho(\mathcal{N}-1)-2\alpha A^{2}(\mathcal{N})b(\mathcal{N})+(\gamma^{2}-\epsilon-2\gamma A(\mathcal{N}))b^{2}(\mathcal{N})\\ &+2(\alpha\gamma-\delta)A(\mathcal{N})b(\mathcal{N})+(\gamma\delta-2\zeta)b(\mathcal{N})+\frac{2}{3}aA^{3}(\mathcal{N})\\ &+(d+\frac{1}{3}a\gamma+\alpha^{2})A^{2}(\mathcal{N})+(\frac{1}{3}a\epsilon+\alpha\delta+2z)A(\mathcal{N}).\end{split}$$		(4.15)

The two equations (4.14), (4.15) are both linear functions of the expression $\Phi(\mathcal{N})$ and $\Phi(\mathcal{N}+1)$ from which $\Phi(\mathcal{N})$ can be found and $\rho(\mathcal{N})$ can be arbitrarily determined for which the corresponding structure function $\Phi(\mathcal{N})$is a polynomial. Hence the solution of the function $\Phi(N)$ which depends on the two parameter u and K is given by two different formulas depending on the value of the parameter $\gamma$.

For $\gamma\neq 0$ we have
	$$\displaystyle\rho(\mathcal{N})=\frac{1}{3.2^{12}.\gamma^{8}(\mathcal{N}+u)(\mathcal{N}+u+1)(1+2(\mathcal{N}+u))^{2}},$$		(4.17)
and,
	$$\displaystyle\begin{split}\Phi(\mathcal{N})={}&-3072\gamma^{6}K(-1+2(\mathcal{N}+u))^{2}\\ &-48\gamma^{6}(\alpha^{2}\epsilon-\alpha\delta\gamma+a\epsilon\gamma-d\gamma^{2})(-3+2(\mathcal{N}+u))(-1+2(\mathcal{N}+u))^{4}(1+2(\mathcal{N}+u))\\ &+\gamma^{8}(-3+2(\mathcal{N}+u))^{2}(-1+2(\mathcal{N}+u))^{4}(1+2(\mathcal{N}+u))^{2}\\ &+768(\alpha\epsilon^{2}-2\delta\epsilon\gamma+4\gamma^{2}\zeta)^{2}+32\gamma^{4}(-1+2(\mathcal{N}+u))^{2}(-1+12(\mathcal{N}+u)\\ &+12(\mathcal{N}+u)^{2})(3\alpha^{2}\epsilon^{2}-6\alpha\delta\epsilon\gamma+2a\epsilon^{2}\gamma+2\delta^{2}\gamma^{2}-4d\epsilon\gamma^{2}+8\gamma^{3}z+4\alpha\gamma^{2}\zeta)\\ &-256\gamma^{2}(-1+2(\mathcal{N}+u))^{2}(3\alpha^{2}\epsilon^{3}-9\alpha\delta\epsilon^{2}\gamma+a\epsilon^{3}\gamma+6\delta^{2}\epsilon\gamma^{2}-3d\epsilon^{2}\gamma^{2}\\ &+2\delta^{2}\gamma^{4}+2d\epsilon\gamma^{4}+12\epsilon\gamma^{3}z-4\gamma^{5}z+12\alpha\epsilon\gamma^{2}\zeta-12\delta\gamma^{3}\zeta+4\alpha\gamma^{4}\zeta).\end{split}$$		(4.18)

The corresponding formulas for $\gamma=0$ and $\epsilon\neq 0$ are given in reference [Daskaloyannis, 2001]. The Casimir operator $K$ can be expressed in terms of the Hamiltonian H alone. An energy dependent Fock space exists if

$\displaystyle\Phi(p+1,u,E)=0,\qquad\Phi(0,u,E)=0,\qquad\Phi(x)>0,$

(4.19)

for any positive integer $x=1,2...p$. The Fock space is then defined by

	$\displaystyle H|E,n>=E|E,n>,\qquad\mathcal{N}|E,n>=n|E,n>,\qquad b|E,0>=0,$
	$\displaystyle b^{\dagger}|n>=\sqrt{\Phi(n+1,E)}|E,n+1>,\qquad b|n>=\sqrt{\Phi(n,E)}|E,n-1>.$		(4.20)

The system given by (4.19) represents a finite-dimensional unitary representation of the dimension p+1. The solutions of the parameter u and energy E corresponding to the representation of the parafermionic algebra of dimension $p+1$ are determined by (4.19).

The 2D quantum Hamiltonian for the harmonic oscillator on the sphere and the hyperbolic plane is given by

$\displaystyle H=\frac{1}{2}\left[\hat{P_{x}^{2}}+\hat{P_{y}^{2}}+\kappa\hat{J^{2}}\right]+\frac{\omega^{2}}{2}\left(\frac{r^{2}}{1-\kappa r^{2}}\right),$

(5.1)

where $r^{2}=x^{2}+y^{2}$. Here, $\hat{P_{x}}$, $\hat{P_{y}}$, $\hat{J}$ are the quantum version of the Noether momenta arising from the three symmetries of the kinetic energy term. They are given by

	$\displaystyle\hat{P_{x}}=-\iota\hslash\sqrt{1-\kappa r^{2}}\frac{\partial}{\partial x},$		(5.2)
	$\displaystyle\hat{P_{y}}=-\iota\hslash\sqrt{1-\kappa r^{2}}\frac{\partial}{\partial y},$		(5.3)
	$\displaystyle\hat{J}=-\iota\hslash\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right).$		(5.4)

It is possible to decompose the Hamiltonian into the following form

	$\displaystyle\hat{H}=$	$\displaystyle\hat{H_{1}}+\hat{H_{2}}+\kappa\hat{J_{12}}^{2},$		(5.5)
where
	$\displaystyle\hat{H_{1}}=$	$\displaystyle\frac{1}{2}\left(\hat{P_{x}}^{2}+\omega^{2}\frac{x^{2}}{1-\kappa r^{2}}\right),$		(5.6)
	$\displaystyle\hat{H_{2}}=$	$\displaystyle\frac{1}{2}\left(\hat{P_{y}}^{2}+\omega^{2}\frac{y^{2}}{1-\kappa r^{2}}\right),$		(5.7)
	$\displaystyle\hat{J_{12}}^{2}=$	$\displaystyle-\frac{\hslash^{2}}{2}\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right).$		(5.8)

It is worth noting that the Hamiltonian Ĥ commutes with each of these three terms for any value of $\kappa$

$\displaystyle[\hat{H},\hat{H_{1}}]=[\hat{H},\hat{H_{2}}]=[\hat{H},\hat{J_{12}}]=0.$

(5.9)

The vanishing commutators symbolize that the $\kappa$ dependent Hamiltonian is a quantum superintegrable system. Let us take two independent motion integral operators

	$\displaystyle A$	$\displaystyle=2\hat{H_{1}}=\left(\hat{P_{x}}^{2}+\omega^{2}\frac{x^{2}}{1-\kappa r^{2}}\right),$		(5.10)
	$\displaystyle B$	$\displaystyle=2\hat{J_{12}}^{2}=-\hslash^{2}\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right).$		(5.11)

We can now construct $C$ from $A$ and $B$. Let us consider the quantum superintegrable system under consideration for which we can verify the quadratic associative algebra

	$\displaystyle[A,B]=$	$\displaystyle C,$		(5.12)
	$\displaystyle[A,C]=$	$\displaystyle 8\hslash^{2}A^{2}+8\hslash^{2}\kappa\{A,B\}-8\hslash^{2}(2H+\hslash^{2}\kappa)A+16\hslash^{2}(\omega^{2}-\hslash^{2}\kappa^{2})B$
		$\displaystyle+(16\hslash^{4}\kappa H+8\hslash^{4}\omega^{2}),$		(5.13)
	$\displaystyle[B,C]=$	$\displaystyle-8\hslash^{2}\kappa B^{2}-8\hslash^{2}\{A,B\}+16\hslash^{4}A+8\hslash^{2}(2H+\hslash^{2}\kappa)B-16\hslash^{4}H.$		(5.14)

The Casimir operator for this algebra as determined from (4.4) can be expressed in terms of Hamiltonian alone as

$\displaystyle K=-48\hslash^{4}H^{2}-32\hslash^{6}\omega^{2}-96\hslash^{6}\kappa H.$

(5.15)

The structure function of the associated deformed oscillator can be written as the product of four terms

$\displaystyle\Phi(x,u,E)=3221225472\hbar^{12}\Phi_{1}(x,u)\Phi_{2}(x,u,E)\Phi_{3}(x,u)\Phi_{4}(x,u,E),$

(5.16)

where the factor functions are

	$\displaystyle\Phi_{1}(x,u)$	$\displaystyle=-\omega^{2}-2\kappa^{2}\hbar^{2}(x+u)+4\kappa^{2}\hbar^{2}(x+u)^{2},$		(5.17)
	$\displaystyle\Phi_{2}(x,u,E)$	$\displaystyle=-2E\kappa-\omega^{2}-2\kappa^{2}\hbar^{2}(x+u)+4\kappa^{2}\hbar^{2}(x+u)^{2},$		(5.18)
	$\displaystyle\Phi_{3}(x,u)$	$\displaystyle=-\omega^{2}+2\kappa^{2}\hbar^{2}-6\kappa^{2}\hbar^{2}(x+u)+4\kappa^{2}\hbar^{2}(x+u)^{2},$		(5.19)
	$\displaystyle\Phi_{4}(x,u,E)$	$\displaystyle=-2E\kappa-\omega^{2}+2\kappa^{2}\hbar^{2}-6\kappa^{2}\hbar^{2}(x+u)+4\kappa^{2}\hbar^{2}(x+u)^{2}.$		(5.20)

The existence of a finite-dimensional unitary representation of the quadratic algebra of dimension $p+1$ is equivalent to the restrictions (4.19) on the annihilation of the structure function for $x=0$ and $x=p+1$, combined with its positivity for $x=1,2...p$. Solving this system of two equations for the two unknowns $E$ and $u$, one obtains the energy eigenvalues. The four energy cases given below arise by using the formulas $\Phi_{1}(0,u)=0$ or $\Phi_{3}(0,u)=0$ for determining the unknown parameter $u$ and $\Phi_{2}(p+1,u,E)=0$ or $\Phi_{4}(p+1,u,E)=0$ for determining the energy eigenvalue E. In this way, we obtain the values of the parameter $u$

	$\displaystyle u_{1}$	$\displaystyle=\frac{1}{4}\pm\frac{1}{4}\sqrt{1+\frac{4\omega^{2}}{\kappa^{2}\hbar^{2}}},$		(5.21)
	$\displaystyle u_{3}$	$\displaystyle=u_{1}+\frac{1}{2}.$		(5.22)

Note that the appropriate root with positive sign (negative sign) is applicable when $4u_{1}$ is greater (smaller) than one. Similarly, when $4u_{3}$ is greater(smaller) than three then appropriate positive (negative) sign is chosen. The values of the energy eigenvalues are obtained from

	$\displaystyle p+1+u$	$\displaystyle=\frac{1}{4}\pm\frac{1}{4}\sqrt{\frac{8E}{\hbar^{2}\kappa}+1+\frac{4\omega^{2}}{\hbar^{2}\kappa^{2}}},$		(5.23)
	$\displaystyle p+1+u$	$\displaystyle=\frac{3}{4}\pm\frac{1}{4}\sqrt{\frac{8E}{\hbar^{2}\kappa}+1+\frac{4\omega^{2}}{\hbar^{2}\kappa^{2}}}.$		(5.24)

A straightforward tedious calculation shows that

$\displaystyle E_{n}$

$\displaystyle=\frac{\hbar^{2}\kappa}{2}\left(n+1\right)\left(n+4u_{1}\right).$

(5.25)

The relation between $n$ and $p$ is given in Table 1 in all the four cases considered above along with the expressions for the factor functions $\Phi_{i}(i=1,2,3,4)$. As a parenthetical remark, the relation between $n$ and $p$ can also be understood if we substitute the formula (5.25) for $E_{n}$ in (5.23) and (5.24). The quantity under the radical sign becomes a perfect square

$\displaystyle\sqrt{\frac{8E_{n}}{\hbar^{2}\kappa}+1+\frac{4\omega^{2}}{\hbar^{2}\kappa^{2}}}=(2n+4u_{1}+1)^{2}.$

(5.26)

Further discussion of the structure function which provides the energy spectra is given in the following section.

Following algebraic considerations analogous to those given in reference [Daskaloyannis, 2001], we have obtained the energy eigenvalues $E_{n}$ and the value of the parameter $u$ from the requirement that $\Phi_{1}$ or $\Phi_{3}$ vanishes for $x=0$ and $\Phi_{2}$ or $\Phi_{4}$ vanishes for $x=p+1$. Our results are summarized in Table 1.

The energy eigenvalues are found to be given in terms of the value $u_{1}$ of the parameter $u$ according to the formula (5.25)

$$E_{n}=\frac{\hbar^{2}\kappa}{2}(n+1)(n+4u_{1}).$$

It may be recalled that $u_{1}$ is related to $\omega$ according to the formula (5.21) as

$$u_{1}=\frac{1}{4}\pm\frac{1}{4}\sqrt{1+\frac{4\omega^{2}}{\hbar^{2}\kappa^{2}}}.$$

The positive (negative) sign is applicable when $u_{1}$ is greater (smaller) than $1/4$. Now $u_{1}$ is the root of the quadratic equation

$$\omega^{2}=2\kappa\hbar u_{1}\left(2\kappa\hbar u_{1}-\kappa\hbar\right),$$

(6.1)

which results from the requirement that $\Phi_{1}$ given by (5.17) vanishes for $x=0$. In order to see that our results agree with the known results from analytic considerations we introduce the parameter

$\displaystyle\beta=2\kappa\hbar u_{1}.$

(6.2)

From (6.1) and (6.2) we see that

$\displaystyle\omega^{2}=\beta(\beta-\kappa\hbar).$

(6.3)

This is the condition for quantum solubility given in references [Cariñena, Ranada, and Santander, 2007a; Carinena, Ranada, and Santander, 2007; Quesne, 2015] and [Cariñena, Ranada, and Santander, 2004]. Note that

$\displaystyle\lim_{\kappa\to 0}\omega^{2}=\beta^{2}.$

(6.4)

In other words, $\beta$ is the frequency of the flat space isotropic oscillator. In terms of $\beta$, the energy eigenvalues are given byCariñena, Ranada, and Santander (2007a); Carinena, Ranada, and Santander (2007); Quesne (2015)

$\displaystyle E_{n}=\frac{\hbar^{2}\kappa}{2}(n+1)\left(n+\frac{2\beta}{\kappa\hbar}\right).$

(6.5)

We now see that the Euclidean limit of (6.5) correctly reproduces the known results for energy eigenvalues i.e. $(n+1)\hbar\beta$. The expression for the structure function for the deformed oscillator in terms of the parameter $\beta$ is given by

$\displaystyle\Phi_{n}(x,\kappa,\frac{\beta}{\kappa\hbar})=$

$\displaystyle\hbar^{4}\kappa^{4}P(P-1)Q(Q-1)R(R-1)S(S-1),$

(6.6)

where Table 2 gives the expressions for $P$, $Q$, $R$ and $S$ in the four cases considered by us.