Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

The Stäckel 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...

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Дата:	2011
Автори:	Ballesteros, A., Enciso, A., Herranz, F.J., Ragnisco, O., Riglioni, D.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform / A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, D. Riglioni // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 48 назв. — англ.

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spelling	irk-123456789-1471722019-02-14T01:25:11Z Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. The Stäckel 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. 2011 Article Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform / A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, D. Riglioni // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J35; 70H06; 81R12 DOI:10.3842/SIGMA.2011.048 http://dspace.nbuv.gov.ua/handle/123456789/147172 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The Stäckel 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.
format	Article
author	Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D.
spellingShingle	Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D.
author_sort	Ballesteros, A.
title	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
title_short	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
title_full	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
title_fullStr	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
title_full_unstemmed	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
title_sort	superintegrable oscillator and kepler systems on spaces of nonconstant curvature via the stäckel transform
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147172
citation_txt	Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform / A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, D. Riglioni // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 48 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-11T01:31:57Z
last_indexed	2025-07-11T01:31:57Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 048, 15 pages Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform? Ángel BALLESTEROS †, Alberto ENCISO ‡, Francisco J. HERRANZ †, Orlando RAGNISCO § and Danilo RIGLIONI § † Departamento de F́ısica, Universidad de Burgos, E-09001 Burgos, Spain E-mail: angelb@ubu.es, fjherranz@ubu.es ‡ Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M), Consejo Superior de Investigaciones Cient́ıficas, C/ Nicolás Cabrera 14-16, E-28049 Madrid, Spain E-mail: aenciso@icmat.es § Dipartimento di Fisica, Università di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy E-mail: ragnisco@fis.uniroma3.it, riglioni@fis.uniroma3.it Received March 18, 2011, in final form May 12, 2011; Published online May 14, 2011 doi:10.3842/SIGMA.2011.048 Abstract. The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler–Coloumb potentials, in order to obtain ma- ximally superintegrable classical systems on N -dimensional Riemannian spaces of noncon- stant 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. Key words: coupling constant metamorphosis; integrable systems; curvature; harmonic oscil- lator; Kepler–Coulomb; Fradkin tensor; Laplace–Runge–Lenz vector; Taub-NUT; Darboux surfaces 2010 Mathematics Subject Classification: 37J35; 70H06; 81R12 1 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 ∈ RN with canonical Poisson bracket ?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html mailto:angelb@ubu.es mailto:fjherranz@ubu.es mailto:aenciso@icmat.es mailto:ragnisco@fis.uniroma3.it mailto:riglioni@fis.uniroma3.it http://dx.doi.org/10.3842/SIGMA.2011.048 http://www.emis.de/journals/SIGMA/S4.html 2 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni {qi, pj} = δij and the notation: q2 = N∑ i=1 q2i , p2 = N∑ i=1 p2i , \|q\| = √ q2. Let H be an “initial” Hamiltonian, HU an “intermediate” one and H̃ the “final” system given by H = p2 µ(q) + V (q), HU = p2 µ(q) + U(q), H̃ = H U = p2 µ̃(q) + Ṽ (q), (1.1) such that µ̃ = µU and Ṽ = V/U . Then, each second-order integral of motion (symmetry) S of H leads to a new one S̃ corresponding to H̃ through an “intermediate” symmetry SU of HU . In particular, if S and SU are written as S = N∑ i,j=1 aij(q)pipj +W (q) = S0 +W (q), SU = S0 +WU (q), (1.2) then one gets a second-order symmetry of H̃ in the form S̃ = S0 −WUH̃. (1.3) The aim of this paper is to apply the above procedure when the “initial” Hamiltonian H is the N -dimensional (ND) free Euclidean motion, and when HU 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 systems H̃ that are defined on Riemannian spaces of nonconstant curvature. Moreover, we will show that the new potential Ṽ 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 HU is chosen to be the harmonic oscillator we obtain two different final MS Hamiltonians, for which H̃ 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 HU 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 ND 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 per- spective. 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. Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 3 2 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 ND Euclidean space EN . As the “initial” Hamiltonian H (1.1) for the Stäckel procedure we consider the one defining the geodesic motion on EN plus a relevant constant α: H = 1 2 p2 + α. (2.1) 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. Proposition 1. (i) The Hamiltonian (2.1) is endowed with the following constants of motion. • (2N − 3) angular momentum integrals (m = 2, . . . , N): S(m) = ∑ 1≤i<j≤m (qipj − qjpi)2, S(m) = ∑ N−m<i<j≤N (qipj − qjpi)2, S(N) = S(N) ≡ L2, (2.2) where L2 is the square of the total angular momentum. • N2 integrals which are the “seeds” of the Fradkin tensor (i, j = 1, . . . , N): Sij = pipj such that N∑ i=1 Sii = 2(H − α). (2.3) • N integrals which are the “seeds” of the components of the LRL vector (i = 1, . . . , N): Si = N∑ k=1 pk (qkpi − qipk) such that N∑ i=1 S2 i = 2L2(H − α). (2.4) (ii) Each of the three sets {H,S(m)}, {H,S(m)} (m = 2, . . . , N) and {Sii} (i = 1, . . . , N) is formed by N functionally independent functions in involution. (iii) Both sets {H,S(m), S(m), Sii} and {H,S(m), S(m), Si} (m = 2, . . . , N and a fixed index i) are constituted by 2N − 1 functionally independent functions. As “intermediate” Hamiltonians HU (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 SU,(m) ≡ S(m) in (1.2). We recall that, in fact, the spherical symmetry of a central potential on EN 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. 4 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni Proposition 2. (i) The harmonic oscillator Hamiltonian defined by HU = 1 2 p2 + βq2 + γ (2.5) has the (2N−3) angular momentum integrals (2.2) together with N2 additional ones given by the components of the ND Fradkin tensor (i, j = 1, . . . , N): SU,ij = pipj + 2βqiqj such that N∑ i=1 SU,ii = 2(HU − γ). (ii) Each of the three sets {HU , S (m)}, {HU , S(m)} (m = 2, . . . , N) and {SU,ii} (i = 1, . . . , N) is formed by N functionally independent functions in involution. (iii) The set {HU , S (m), S(m), SU,ii} (m = 2, . . . , N and a fixed index i) provides 2N − 1 func- tionally independent functions. Proposition 3. (i) The KC Hamiltonian given by HU = 1 2 p2 + δ \|q\| + ξ (2.6) has the (2N −3) angular momentum integrals (2.2) together with the N components of the LRL vector (i = 1, . . . , N): SU,i = N∑ k=1 pk (qkpi − qipk)− δqi \|q\| such that N∑ i=1 S2 U,i = 2L2(HU − ξ) + δ2. (ii) Each of the two sets {HU , S (m)} and {HU , S(m)} (m = 2, . . . , N) is formed by N func- tionally independent functions in involution. (iii) The set {HU , S (m), S(m), SU,i} (m = 2, . . . , N and a fixed index i) is constituted by 2N − 1 functionally independent functions. 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 = ω2/2 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”. 3 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 HU (2.5), then we obtain the final Hamiltonian H̃ H̃ = p2 2(γ + βq2) + α γ + βq2 , (3.1) Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 5 so that the relations (1.1) read as µ = 2, V = α, U = γ + βq2, µ̃ = 2 ( γ + βq2 ) , Ṽ = α γ + βq2 . As far as the symmetries S = S0 +W (1.2) are concerned, we find from Proposition 1 that S (m) 0 = S(m), W (m) = 0, S0,(m) = S(m), W(m) = 0, S0,ij = Sij , Wij = 0, while from Proposition 2 we obtain the elements WU for the decompositions of SU = S0 +WU , W (m) U = 0, WU,(m) = 0, WU,ij = 2βqiqj , where m = 2, . . . , N and i, j = 1, . . . , N . Consequently, the Hamiltonian H̃ (3.1) is Stäckel equivalent to the free Euclidean motion, through the harmonic oscillator potential, and its integrals of motion S̃ come from (1.3) and turn out to be S̃(m) = S(m), S̃(m) = S(m), S̃ij = pipj − 2βqiqjH̃(q,p). (3.2) Thus we have obtained the (2N − 3) angular momentum integrals S(m) and S(m), together with N2 ones, S̃ij , which form a curved Fradkin tensor. The quasi-MS property of H̃ is ensured by the preservation of the (2N − 3) angular momentum integrals, that is, each of the two sets {H̃, S(m)}, {H̃, S(m)} (m = 2, . . . , N) is formed by N functionally independent functions in involution. Hence, from now on, we assume this fact and only pay attention to the additional constants S̃ij which characterize (3.1) as a MS system. In order to perform a preliminary geometrical analysis of H̃ we recall that, in general, any Hamiltonian of the form H = p2 2f(\|q\|)2 + V(\|q\|) can be interpreted as describing a particle (with unit mass) on an ND 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] ds2 = f(\|q\|)2dq2, R = −(N − 1) ( (N − 4)f ′(r)2 + f(r) ( 2f ′′(r) + 2(N − 1)r−1f ′(r) ) f(r)4 ) , (3.3) 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: UKC(r) := ∫ r dr′ r′2f(r′) , UO(r) := 1 UKC(r)2 , (3.4) that was introduced in [14] up to additive and multiplicative constants. With these ideas in mind, we now analyze the specific systems defined by H̃ (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 otherwise H̃ is again the initial H. Therefore we are led to consider two different cases with generic α: (i) β 6= 0, γ = 0; and (ii) β 6= 0, γ 6= 0. 6 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni 3.1 The case with β 6= 0 and γ = 0: a curved hyperbolic KC system If γ = 0 we scale H̃ to deal with the Hamiltonian HKC = βH̃ = p2 2q2 + α q2 . (3.5) Then, f(\|q\|) = \|q\| = r so the metric and scalar curvature (3.3) on M reduces to ds2 = q2dq2, R = −3(N − 1)(N − 2) r4 , (3.6) while the intrinsic KC and oscillator potentials (3.4) on this curved space would be UKC(r) = − 1 2r2 , UO(r) = 4r4. The latter result shows that HKC (3.5) always defines an intrinsic KC potential on the spaceM. 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 toHKC (3.5) the Kustaanheimo– Stiefel canonical transformation defined by [39, 40] q̃1 = 1 2 ( q21 − q22 ) , p̃1 = p1q1 − p2q2 q21 + q22 , q̃2 = q1q2, p̃2 = p2q1 + p1q2 q21 + q22 , (3.7) so with canonical Poisson bracket {q̃i, p̃j} = δij . In this way we recover the 2D KC Hamiltonian HKC = p21 + p22 2(q21 + q22) + α q21 + q22 = 1 2 ( p̃21 + p̃22 ) + α 2 √ q̃21 + q̃22 and the five symmetries (3.2) reduce to three integrals of motion, namely S(2) = S(2) = L2 = (q1p2 − q2p1)2 = 4(q̃1p̃2 − q̃2p̃1)2, S̃11 = p21 − 2q21HKC = 2p̃2(q̃2p̃1 − q̃1p̃2)− αq̃1√ q̃21 + q̃22 − α, S̃22 = −S̃11 − 2α, S̃12 = S̃21 = p1p2 − 2q1q2HKC = 2p̃1(q̃1p̃2 − q̃2p̃1)− αq̃2√ q̃21 + q̃22 . 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 com- ing from the 2D Fradkin tensor reduce to the two components of the LRL vector: (S̃11, S̃22)→ S̃1 and (S̃12, S̃21)→ S̃2. Consequently, a proper curved KC system arises whenever N ≥ 3 and its full integrability properties can be summarized as follows. Proposition 4. (i) For N ≥ 3, the Hamiltonian HKC (3.5) determines an intrinsic KC system on the hyper- bolic space of nonconstant curvature (3.6). (ii) HKC is endowed with the (2N − 3) angular momentum integrals (2.2) together with N2 ones which are the components of an ND curved Fradkin tensor (i, j = 1, . . . , N): S̃ij = pipj − 2qiqjHKC such that N∑ i=1 S̃ii = −2α and {S̃ii, S̃jj} = 0. (iii) The set {HKC, S (m), S(m), S̃ii} (m = 2, . . . , N and a fixed index i) is formed by 2N − 1 functionally independent functions. Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 7 3.2 The case with β 6= 0 and γ 6= 0: the Darboux III oscillator If both β, γ 6= 0, we can write the Hamiltonian (3.1) in the form Hλ = γH̃ − α = p2 2(1 + λq2) − λαq2 1 + λq2 , λ = β/γ. (3.8) Then the metric and scalar curvature (3.3) on the corresponding manifold M are given by ds2 = ( 1 + λq2 ) dq2, R = −λ (N − 1) ( 2N + 3(N − 2)λr2 ) (1 + λr2)3 , (3.9) and the intrinsic potentials (3.4) read UKC(r) = − √ 1 + λr2 r , UO(r) = r2 1 + λr2 . (3.10) In this way, we recover the ND 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]: λ > 0 : R(0) = −2λN(N − 1), r ∈ [0,∞); λ < 0 : R(0) = 2\|λ\|N(N − 1), r ∈ [0, 1/ √ \|λ\|), 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 ND hyperbolic space with negative constant sectional curvature equal to −2λ for λ > 0, or with that corresponding to the ND 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 Hλ 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 ω2 = −2λα and, in that form, the limit λ→ 0 gives rise to the harmonic oscillator on EN , so λ behaves as a classical deformation parameter governing the curvature and the potential. The MS property of Hλ is then characterized by [13, 15]: Proposition 5. (i) The Hamiltonian Hλ (3.8) defines an intrinsic curved hyperbolic Darboux oscillator for λ > 0 and r ∈ [0,∞) and a curved spherical Darboux one for λ < 0 and r ∈ [0, 1/ √ \|λ\|). (ii) Besides the (2N−3) angular momentum integrals (2.2), Hλ Poisson-commutes with the N2 components of the ND curved Fradkin tensor (i, j = 1, . . . , N) given by S̃ij = pipj − 2λqiqj (Hλ + α) such that N∑ i=1 S̃ii = 2Hλ and {S̃ii, S̃jj} = 0. (iii) The set {Hλ, S(m), S(m), S̃ii} (m = 2, . . . , N and a fixed index i) is formed by 2N − 1 functionally independent functions. 4 Superintegrable systems from the Kepler–Coulomb potential In this case we consider the initial Hamiltonian H (2.1) and the KC system HU (2.6) for the intermediate one; this provides the final Hamiltonian H̃ (1.1) H̃ = \|q\|p2 2(δ + ξ\|q\|) + α\|q\| δ + ξ\|q\| , (4.1) 8 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni where µ = 2, V = α, U = δ + ξ\|q\| \|q\| , µ̃ = 2(δ + ξ\|q\|) \|q\| , Ṽ = α\|q\| δ + ξ\|q\| . From Proposition 1 we obtain the decomposition of the symmetries S = S0 +W (2.2) and (2.4) (m = 2, . . . , N and i = 1, . . . , N): S (m) 0 = S(m), W (m) = 0, S0,(m) = S(m), W(m) = 0, S0,i = Si, Wi = 0, while from Proposition 3 we find the one corresponding to SU = S0 +WU (1.2) W (m) U = 0, WU,(m) = 0, WU,i = −δqi \|q\| . Therefore, the Hamiltonian H̃ (4.1) is Stäckel equivalent to the free Euclidean motion, through the KC potential, and its integrals of motion S̃ (1.3) are given by S̃(m) = S(m), S̃(m) = S(m), S̃i = N∑ k=1 pk (qkpi − qipk) + δqi \|q\| H̃(q,p). (4.2) Hence, H̃ (4.1) is endowed with the (2N − 3) angular momentum integrals (2.2) together with a curved LRL N -vector with components S̃i. Notice that the parameter δ cannot vanish in order to avoid the initial system H. Thus, similarly to the previous section, we study two systems covered by H̃ (4.1): (i) δ 6= 0, ξ = 0; and (ii) δ 6= 0, ξ 6= 0. 4.1 The case with δ 6= 0 and ξ = 0: a curved spherical oscillator system If ξ = 0 we have the Hamiltonian system HO = δH̃ = 1 2 \|q\|p2 + α\|q\|. (4.3) We stress that for N = 3 this system was early considered in [2]. The metric and scalar curvature (3.3) give ds2 = 1 \|q\| dq2, R = 3(N − 1)(N − 2) 4r , (4.4) and the intrinsic KC and oscillator potentials (3.4) turn out to be UKC(r) = − 2√ r , UO(r) = r 4 . Hence HO (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 transformation q̃1 = q2(√ q21 + q22 − q1 )1/2 , p̃1 = (p1q2 − 2p2q1) (√ q21 + q22 − q1 ) + p2q 2 2(√ q21 + q22 − q1 )3/2 , q̃2 = (√ q21 + q22 − q1 )1/2 , p̃2 = p2q2 − p1 (√ q21 + q22 − q1 ) (√ q21 + q22 − q1 )1/2 , Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 9 which is just the inverse of the canonical transformation (3.7); this yields the expected system HO = 1 2 √ q21 + q22 ( p21 + p22 ) + α √ q21 + q22 = 1 4 ( p̃21 + p̃22 ) + 1 2 α ( q̃21 + q̃22 ) . The canonical transformation of the three symmetries (4.2) gives S(2) = S(2) = L2 = (q1p2 − q2p1)2 = 1 4 (q̃1p̃2 − q̃2p̃1)2, S̃1 = p2(q2p1 − q1p2) + q1√ q21 + q22 HO = 1 4 ( p̃21 − p̃22 ) + 1 2 α ( q̃21 − q̃22 ) , S̃2 = p1(q1p2 − q2p1) + q2√ q21 + q22 HO = 1 2 p̃1p̃2 + αq̃1q̃2. Then the four components of the 2D Euclidean Fradkin tensor S̃ij are recovered, in the new canonical variables, from the set of constants (HO, S̃1, S̃2) by setting S̃11 = 2(HO + S̃1) = p̃21 + 2αq̃21, S̃22 = 2(HO − S̃1) = p̃22 + 2αq̃22, S̃12 = S̃21 = 2S̃2 = p̃1p̃2 + 2αq̃1q̃2. Therefore the proper curved system arises whenever N ≥ 3, which yields the following Proposition 6. (i) For N ≥ 3, the Hamiltonian HO (4.3) defines an intrinsic oscillator potential on the spherical space of nonconstant curvature (4.4). (ii) HO Poisson-commutes with the (2N − 3) angular momentum integrals (2.2) and with the components of the LRL N -vector given by (i = 1, . . . , N): S̃i = N∑ k=1 pk (qkpi − qipk) + qi \|q\| HO such that N∑ i=1 S̃2 i = H2 O − 2αL2. (iii) The set {HO, S (m), S(m), S̃i} (m = 2, . . . , N and a fixed index i) is formed by 2N − 1 functionally independent functions. 4.2 The case with δ 6= 0 and ξ 6= 0: the Taub-NUT oscillator We scale the Hamiltonian (4.1) as Hη = ξH̃ = \|q\|p2 2(η + \|q\|) + α\|q\| η + \|q\| , η = δ/ξ. (4.5) 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 ds2 = η + \|q\| \|q\| dq2, R = η(N − 1) ( 4(N − 3)r + 3(N − 2)η ) 4r(η + r)3 , (4.6) so that the domain of r = \|q\| in M depends on the sign of η: η > 0 : r ∈ (0,∞); η < 0 : r ∈ [\|η\|,∞). (4.7) 10 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni The intrinsic potentials (3.4) are given by UKC(r) = −2 η √ η + r r , UO(r) = η2r 4(η + r) . (4.8) Consequently, Hη 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]: HTaub-NUT = p2 2(1 + 4m/\|q\|) + µ2 2(4m)2 ( 1 + 4m \|q\| ) = \|q\|p2 2(4m+ \|q\|) + µ2\|q\|/(4m)2 2(4m+ \|q\|) + µ2 2\|q\|(4m+ \|q\|) + µ2/(4m) 4m+ \|q\| . (4.9) The relationship with Hη is established by setting η = 4m, α = − µ2 2(4m)2 , which gives Hη=4m + µ2 (4m)2 = \|q\|p2 2(4m+ \|q\|) + µ2\|q\|/(4m)2 2(4m+ \|q\|) + µ2/(4m) 4m+ \|q\| , so that we recover three terms in the “expanded” expression for HTaub-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 hyper- spherical coordinates in the form [14] p2 = p2r + r−2L2 and next L2 → L2 + µ2. From this viewpoint, Hη can be regarded as an ND 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 Hη are summarized in Proposition 7. (i) The Hamiltonian Hη (4.5) characterizes two intrinsic oscillator potentials on the corre- sponding Riemannian space of nonconstant curvature (4.6) according to (4.7). (ii) Hη is endowed with the (2N − 3) angular momentum integrals (2.2) together with the components of the curved LRL N -vector given by (i = 1, . . . , N): S̃i = N∑ k=1 pk (qkpi − qipk) + η qi \|q\| Hη such that N∑ i=1 S̃2 i = 2L2(Hη − α) + η2H2 η. (iii) The set {Hη, S(m), S(m), S̃i} (m = 2, . . . , N and a fixed index i) is formed by 2N − 1 functionally independent functions. Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 11 Table 1. Maximally superintegrable classical oscillator and KC Hamiltonians in N dimensions. • Geodesic motion on Euclidean space H = 1 2 p2 + α ∗ Common (2N − 3) angular momentum integrals of motion S(m) = ∑ 1≤i<j≤m (qipj − qjpi)2, S(m) = ∑ N−m<i<j≤N (qipj − qjpi)2 , S(N) = S(N) ≡ L2 ∗ “Seeds” of the ND Fradkin tensor ∗ “Seeds” of the LRL N -vector Sij = pipj Si = N∑ k=1 pk (qkpi − qipk) • Harmonic oscillator • Euclidean KC HU = 1 2 p2 + βq2 + γ HU = 1 2 p2 + δ \|q\| + ξ ∗ Flat ND Fradkin tensor ∗ Flat LRL N -vector SU,ij = pipj + 2βqiqj SU,i = N∑ k=1 pk (qkpi − qipk)− δ qi\|q\| • Curved hyperbolic KC (N ≥ 3) • Curved spherical oscillator (N ≥ 3) HKC = p2 2q2 + α q2 HO = 1 2 \|q\|p2 + α\|q\| ∗ Curved ND Fradkin tensor ∗ Curved LRL N -vector S̃ij = pipj − 2qiqjHKC S̃i = N∑ k=1 pk (qkpi − qipk) + qi \|q\|HO • Darboux III oscillator • Taub-NUT oscillator Hλ = p2 2(1 + λq2) − λαq2 1 + λq2 Hη = \|q\|p2 2(η + \|q\|) + α\|q\| η + \|q\| ∗ Curved ND Fradkin tensor ∗ Curved LRL N -vector S̃ij = pipj − 2λqiqj (Hλ + α) S̃i = N∑ k=1 pk (qkpi − qipk) + η qi \|q\|Hη 5 Outlook and superintegrable quantization So far we have obtained and interpreted four MS classical Hamiltonian systems on Riemannian spaces of nonconstant curvature by starting from free motion on EN and applying the Stäckel transform through the harmonic oscillator and KC potentials. The main results here obtained are displayed in 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 Jij = qipj − qjpi with i < j and i, j = 1, . . . , N which span the so(N) Lie–Poisson algebra {Jij , Jik} = Jjk, {Jij , Jjk} = −Jik, {Jik, Jjk} = Jij , i < j < k. Then the “common” (2N − 3) angular momentum integrals S(m) and S(m) (2.2) can be written 12 Á. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni Table 2. Maximally superintegrable quantum oscillator and KC Hamiltonians in N dimensions. ∗ Common (2N − 3) quantum angular momentum operators Ŝ(m) = ∑ 1≤i<j≤m (q̂ip̂j − q̂j p̂i)2, Ŝ(m) = ∑ N−m<i<j≤N (q̂ip̂j − q̂j p̂i)2 , Ŝ(N) = Ŝ(N) ≡ L̂2 • Quantum hyperbolic KC (N ≥ 3) • Quantum spherical oscillator (N ≥ 3) ĤKC = 1 2q̂2 p̂2 + α q̂2 ĤO = 1 2 \|q̂\|p̂2 + α\|q̂\| ∗ Quantum ND Fradkin tensor ∗ Quantum LRL N -vector Ŝij = p̂ip̂j − 2q̂iq̂jĤKC Ŝi = 1 2 N∑ k=1 p̂k (q̂kp̂i − q̂ip̂k) + 1 2 N∑ k=1 (q̂kp̂i − q̂ip̂k) p̂k + q̂i \|q̂\| ĤO N∑ i=1 Ŝii = −2α N∑ i=1 Ŝ2 i = Ĥ2 O − 2αL̂2 − 1 2 (N − 1)2~2α • Quantum Darboux III oscillator • Quantum Taub-NUT oscillator Ĥλ = 1 2(1 + λq̂2) p̂2 − λα q̂2 1 + λq̂2 Ĥη = \|q̂\| 2(η + \|q̂\|) p̂2 + α\|q̂\| η + \|q̂\| ∗ Quantum ND Fradkin tensor ∗ Quantum LRL N -vector Ŝij = p̂ip̂j − 2λq̂iq̂j ( Ĥλ + α ) Ŝi = 1 2 N∑ k=1 p̂k (q̂kp̂i − q̂ip̂k) + 1 2 N∑ k=1 (q̂kp̂i − q̂ip̂k) p̂k + η q̂i \|q̂\| Ĥη N∑ i=1 Ŝii = 2Ĥλ N∑ i=1 Ŝ2 i = 2L̂2(Ĥη − α) + η2Ĥ2 η + 1 2 (N − 1)2~2(Ĥη − α) as the quadratic Casimirs of some rotation subalgebras so(m) ⊂ so(N): S(m) = ∑ 1≤i<j≤m J2 ij , S(m) = ∑ N−m<i<j≤N J2 ij , with S(N) = S(N) = L2 being the quadratic Casimir of so(N). In this respect, we also notice that all the LRL constants of motion (Si, SU,i, S̃i) given in Table 1 are transformed as N -vectors under the action of the generators of so(N) (as it should be): {Jij , S̃k} = δikS̃j − δjkS̃i. Furthermore, all of these systems possess an sl(2,R) coalgebra symmetry as well [14]. If we denote J− = q2, J+ = p2 and J3 = q · p we have that {J3, J+} = 2J+, {J3, J−} = −2J−, {J−, J+} = 4J3, 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(q2,p2,q · p) so that this always provides, at least, a quasi-MS system [32, 33]. There- fore 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]). Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature 13 Therefore, let us consider the quantum position and momenta operators, q̂, p̂, with canonical Lie bracket [q̂i, p̂j ] = i~δij . The resulting MS quantum Hamiltonians are summarized in the following final statement. Proposition 8. Let H be one of the classical Hamiltonians given in Propositions 4–7. (i) The Schrödinger quantization of H and its quantum symmetries are given in Table 2. (ii) The quantum Hamiltonian Ĥ is endowed with (2N − 3) quantum angular momentum ope- rators Ŝ(m) and Ŝ(m), such that {Ĥ, Ŝ(m)} or {Ĥ, Ŝ(m)} (m = 2, . . . , N) is a set of N algebraically independent commuting observables. (iii) If Ĥ ∈ (ĤKC, Ĥλ) then it commutes with the N2 components, Ŝij, of a quantum Fradkin tensor (i, j = 1, . . . , N). The set {Ĥ, Ŝ(m), Ŝ(m), Ŝii} (m = 2, . . . , N and a fixed index i) is formed by 2N − 1 algebraically independent commuting observables. (iv) When Ĥ ∈ (ĤO, Ĥη) this commutes with the N components, Ŝi, of a quantum LRL vector (i = 1, . . . , N). The set {Ĥ, Ŝ(m), Ŝ(m), Ŝi} (m = 2, . . . , N and a fixed index i) is constituted by 2N − 1 algebraically independent commuting observables. 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Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

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