Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the St\"ackel Transform

The St\"ackel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N-dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.


Introduction
The coupling constant metamorphosis or Stäckel transform was formerly introduced in [1,2] and further developed and applied to several classical and quantum Hamiltonian systems in [3,4,5,6,7]. This approach has proven to be a useful tool in order to relate different (super)integrable systems together with their associated symmetries and to deduce new integrable Hamiltonian systems starting from known ones.
For our purposes, the classical Stäckel transform can be briefly summarized as follows [3,4]. Consider the conjugate coordinates and momenta q, p ∈ R N with canonical Poisson bracket 2Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni {q i , p j } = δ ij and the notation: Let H be an "initial" Hamiltonian, H U an "intermediate" one andH the "final" system given by a ij (q)p i p j + W (q) = S 0 + W (q), then one gets a second-order symmetry ofH in the form 3) The aim of this paper is to apply the above procedure when the "initial" Hamiltonian H is the N -dimensional (N D) free Euclidean motion, and when H U is either the isotropic harmonic oscillator or the Kepler-Coulomb (KC) Hamiltonian. It is well known that these three systems are maximally superintegrable (MS), that is, they are endowed with the maximum number of 2N − 1 functionally independent integrals of motion (in these cases, all of them are quadratic in the momenta). These three systems and their MS property are briefly recalled in the next section, and we will see that the Stäckel transform gives rise to several MS systemsH that are defined on Riemannian spaces of nonconstant curvature. Moreover, we will show that the new potentialṼ can be interpreted as either an (intrinsic) oscillator or a KC potential on the corresponding curved manifold. In this way, by starting from the Euclidean Fradkin tensor [8], formerly studied by Demkov in [9], and the Laplace-Runge-Lenz (LRL) N -vector, the Stäckel transform provides for each case its curved analogue (see [10,11,12] and references therein).
In particular, we show in Section 3 that if H U is chosen to be the harmonic oscillator we obtain two different final MS Hamiltonians, for whichH is endowed with a curved Fradkin tensor; these are a KC system on a hyperbolic space of nonconstant curvature and the so-called Darboux III oscillator [13,14,15]. In Section 4 we take H U as the (flat) KC Hamiltonian and the Stäckel transform leads to other two different MS systems together with their curved LRL N -vector; both of them are interpreted as intrinsic oscillators on curved Riemannian manifolds. Surprisingly enough, one of them is the N D generalization of the Taub-NUT oscillator [16,17,18,19,20,21,22,23]. We stress that for some systems the dimension N = 2 is rather special as the underlying manifold remains flat, meanwhile for N ≥ 3 such systems are defined on proper curved spaces (see Sections 3.1 and 4.1). This is similar to what happens in the classifications of 2D and 3D integrable systems on spaces of constant curvature (including the flat Euclidean one) [24,25,26,27,28,29,30,31] which exhibit some differences according to the dimension and, in general, the 3D case is usually the cornerstone for the generalization of a given system to arbitrary dimension.
As a byproduct of this construction, the "growth" of the Fradkin tensor and the LRL vector from their Euclidean "seeds" to their curved counterparts can be highlighted from a global perspective. These results are comprised in Table 1 in the last section. Furthermore we also present in Table 2 the MS quantization for all of the above systems together with their "additional" quantum Fradkin/LRL symmetries.

Harmonic oscillator and Kepler potentials on Euclidean space
In order to fix a suitable common framework, we briefly recall the well-known basics of the superintegrability properties of the Hamiltonians describing free motion, harmonic oscillator and KC potentials on the N D Euclidean space E N .
As the "initial" Hamiltonian H (1.1) for the Stäckel procedure we consider the one defining the geodesic motion on E N plus a relevant constant α: Obviously this is a MS system, and there are many possibilities to choose its integrals of motion. We shall make use, throughout the paper, of the following results.
where L 2 is the square of the total angular momentum. • N 2 integrals which are the "seeds" of the Fradkin tensor (i, j = 1, . . . , N ): • N integrals which are the "seeds" of the components of the LRL vector (i = 1, . . . , N ): As "intermediate" Hamiltonians H U (1.1) we consider either the harmonic oscillator or the KC system. Since both of them are central potentials, the angular momentum integrals (2.2) are valid for both cases, that is, S (m) U ≡ S (m) and S U,(m) ≡ S (m) in (1.2). We recall that, in fact, the spherical symmetry of a central potential on E N directly provides such (2N − 3) independent angular momentum integrals, so they characterize a quasi-MS system [32,33]. However what makes rather special the harmonic oscillator and KC systems is the existence of one more independent integral, which is extracted from a new set of integrals that ensure their MS property and is related to the fact that these two systems are the only ones fulfilling the classical Bertand's theorem [34]. In this respect, each of the sets of integrals (2.3) and (2.4) gives rise to one known set of additional constants for the harmonic oscillator and KC system, respectively.
has the (2N − 3) angular momentum integrals (2.2) together with N 2 additional ones given by the components of the ND Fradkin tensor (i, j = 1, . . . , N ): is formed by N functionally independent functions in involution.

(i) The KC Hamiltonian given by
has the (2N − 3) angular momentum integrals (2.2) together with the N components of the LRL vector (i = 1, . . . , N ): In the two next sections we apply the Stäckel transform to each of these two MS systems. Notice that the proper isotropic harmonic oscillator arises whenever β = ω 2 /2 with frequency ω and γ = 0, while the Kepler one corresponds to set δ = −K and ξ = 0. We remark that in this approach the constant α is essential in order to obtain a curved potential while the others β, γ, δ and ξ enter in both the kinetic and the potential term giving rise to MS oscillator/KC potentials on Riemannian spaces of nonconstant curvature, so that they can be regarded as classical "deformation parameters".

Superintegrable systems from harmonic oscillator potential
If we consider as the initial Hamiltonian H the free system (2.1) and as the intermediate one the harmonic oscillator H U (2.5), then we obtain the final HamiltonianH so that the relations (1.1) read as As far as the symmetries S = S 0 + W (1.2) are concerned, we find from Proposition 1 that while from Proposition 2 we obtain the elements W U for the decompositions of where m = 2, . . . , N and i, j = 1, . . . , N . Consequently, the HamiltonianH (3.1) is Stäckel equivalent to the free Euclidean motion, through the harmonic oscillator potential, and its integrals of motionS come from (1.3) and turn out to bẽ Hence, from now on, we assume this fact and only pay attention to the additional constantsS ij which characterize (3.1) as a MS system. In order to perform a preliminary geometrical analysis ofH we recall that, in general, any Hamiltonian of the form can be interpreted as describing a particle (with unit mass) on an N D spherically symmetric space M under the action of the central potential V(|q|) [14]. The metric and scalar curvature of M are given by [35] where we have introduced the radial coordinate r = |q|. For general results on 2D and 3D (super)integrable systems on conformally flat spaces we refer to [36,37,38]. Furthermore, the conformal factor f (|q|) = f (r) is directly related, under the following prescription, with the intrinsic KC and oscillator potentials on M: that was introduced in [14] up to additive and multiplicative constants. With these ideas in mind, we now analyze the specific systems defined byH (3.1) according to the values of the parameters β and γ. Notice that α is the constant which governs the potential, so to setting α = 0 leads to geodesic motion on M, and that β must be always different from zero, since otherwiseH is again the initial H. Therefore we are led to consider two different cases with generic α: 3.1 The case with β = 0 and γ = 0: a curved hyperbolic KC system If γ = 0 we scaleH to deal with the Hamiltonian Then, f (|q|) = |q| = r so the metric and scalar curvature (3.3) on M reduces to while the intrinsic KC and oscillator potentials (3.4) on this curved space would be The latter result shows that H KC (3.5) always defines an intrinsic KC potential on the space M. Nevertheless the curvature (3.6) vanishes for N = 2, while the space is of nonconstant curvature for N ≥ 3. Therefore, for N = 2 the Hamiltonian must correspond to the usual KC system on the Euclidean space. This fact can be explicitly proven by applying to H KC (3.5) the Kustaanheimo-Stiefel canonical transformation defined by [39,40] so with canonical Poisson bracket {q i ,p j } = δ ij . In this way we recover the 2D KC Hamiltonian and the five symmetries (3.2) reduce to three integrals of motion, namely Hence, by taking into account Proposition 3 for N = 2, we find that, under the above canonical transformation, the only angular momentum integral S (2) is kept, while the four constants coming from the 2D Fradkin tensor reduce to the two components of the LRL vector: (S 11 ,S 22 ) →S 1 and (S 12 ,S 21 ) →S 2 . Consequently, a proper curved KC system arises whenever N ≥ 3 and its full integrability properties can be summarized as follows.
Then the metric and scalar curvature (3.3) on the corresponding manifold M are given by and the intrinsic potentials (3.4) read (3.10) In this way, we recover the N D spherically symmetric generalization of the Darboux surface of type III [41,42,43,44] introduced in [14,35]. Notice that the domain of r = |q| and the type of the underlying curved manifold depends on the sign of λ [15]: where we have written the value of the scalar curvature (3.9) at the origin r = 0. We stress that R(0) coincides either with the scalar curvature of the N D hyperbolic space with negative constant sectional curvature equal to −2λ for λ > 0, or with that corresponding to the N D spherical space with sectional curvature equal to 2|λ| for λ < 0. By taking into account the above geometrical considerations and expressions (3.10), we find that H λ comprises both an intrinsic hyperbolic oscillator potential and a spherical one on M according to the sign of λ. Strictly speaking the curved oscillator potentials arise by introducing the frequency ω 2 = −2λα and, in that form, the limit λ → 0 gives rise to the harmonic oscillator on E N , so λ behaves as a classical deformation parameter governing the curvature and the potential. The MS property of H λ is then characterized by [13,15] while from Proposition 3 we find the one corresponding to Therefore, the HamiltonianH (4.1) is Stäckel equivalent to the free Euclidean motion, through the KC potential, and its integrals of motionS (1.3) are given bỹ Hence,H (4.1) is endowed with the (2N − 3) angular momentum integrals (2.2) together with a curved LRL N -vector with componentsS i .

4.1
The case with δ = 0 and ξ = 0: a curved spherical oscillator system If ξ = 0 we have the Hamiltonian system We stress that for N = 3 this system was early considered in [2]. The metric and scalar curvature (3.3) give , (4.4) and the intrinsic KC and oscillator potentials (3.4) turn out to be Hence H O (4.3) determines an intrinsic oscillator potential on M. However, for N = 2 the curvature is equal to zero, so this case should actually be the 2D harmonic oscillator. This can be proven by means of the canonical transformatioñ , which is just the inverse of the canonical transformation (3.7); this yields the expected system The canonical transformation of the three symmetries (4.2) gives Then the four components of the 2D Euclidean Fradkin tensorS ij are recovered, in the new canonical variables, from the set of constants (H O ,S 1 ,S 2 ) by setting Therefore the proper curved system arises whenever N ≥ 3, which yields the following Proposition 6.
Notice that the limit η → 0 reduces to the free Hamiltonian in Euclidean space. The metric and scalar curvature (3.3) on the corresponding manifold M turn out to be so that the domain of r = |q| in M depends on the sign of η: 10Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni The intrinsic potentials (3.4) are given by . (4.8) Consequently, H η defines two intrinsic oscillators, which are different systems according to (4.7). It is worth comparing (4.5) with the Taub-NUT system [16,17,18,19,20,21,22,23] which can be written as [14]: The relationship with H η is established by setting so that we recover three terms in the "expanded" expression for H Taub-NUT (4.9); namely, the kinetic term defining the geodesic motion on the Taub-NUT space (4.6), the insintric oscillator potential (4.8) and the one which comes out by adding a constant to the oscillator potential. There is one missing term, the third one in (4.9), which corresponds to the Dirac monopole. However we notice that this can be derived from the angular momentum by introducing hyperspherical coordinates in the form [14] p 2 = p 2 r + r −2 L 2 and next L 2 → L 2 + µ 2 .
From this viewpoint, H η can be regarded as an N D MS generalization of the Taub-NUT system which is recovered for η > 0, being the case with η < 0 a different physical oscillator potential. The symmetry properties for H η are summarized in Proposition 7.   • Geodesic motion on Euclidean space  Table 1 where the transition from the "seeds" of the Fradkin tensor and the LRL vector up to their curved analogues is laid bare by reading the table through its two columns. Recall, however, that the Darboux III and the Taub-NUT oscillators give rise, each of them, to two different physical systems according to the sign of the parameters λ and η, respectively. Some related comments are in order. All the Hamiltonians shown in Table 1 are constructed on spherically symmetric spaces so that they are endowed with an so(N ) Lie-Poisson symmetry. In particular, let us consider the generators of rotations J ij = q i p j − q j p i with i < j and i, j = 1, . . . , N which span the so(N ) Lie-Poisson algebra

) angular momentum integrals of motion
Then the "common" (2N − 3) • Quantum Darboux III oscillator • Quantum Taub-NUT oscillator H λ = 1 2(1 + λq 2 )p 2 − λαq 2 1 + λq 2Ĥ η = |q| 2(η + |q|)p 2 + α|q| η + |q| * Quantum N D Fradkin tensor * Quantum LRL N -vector as the quadratic Casimirs of some rotation subalgebras so(m) ⊂ so(N ): with S (N ) = S (N ) = L 2 being the quadratic Casimir of so(N ). In this respect, we also notice that all the LRL constants of motion (S i , S U,i ,S i ) given in Table 1 are transformed as N -vectors under the action of the generators of so(N ) (as it should be): Furthermore, all of these systems possess an sl(2, R) coalgebra symmetry as well [14]. If we denote J − = q 2 , J + = p 2 and J 3 = q · p we have that and the common integrals (2.2) are just the mth (left and right) coproducts of the Casimir of sl(2, R). This set of (2N − 3) integrals is "universal" for any Hamiltonian function defined by H = H(q 2 , p 2 , q · p) so that this always provides, at least, a quasi-MS system [32,33]. Therefore the Hamiltonians shown in Table 1 are distinguished systems since they have "additional" symmetries.
To end with, we shall present the MS quantization of the four curved classical systems. For this purpose, we remark that the MS quantization of the Darboux III oscillator has been recently obtained in [45], and the corresponding quantum dynamics has been fully solved for λ > 0 (the case with λ < 0 is still an open problem). This quantization has been obtained by applying the so called "Schrödinger quantization" procedure [46], and its relationship with the Laplace-Beltrami and position-dependent-mass quantizations has been established in [47] by means of similarity transformations (see also [48]).