Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

The D-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyp...

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Datum:	2011
1. Verfasser:	Quesne, C.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2011
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1468042019-02-12T01:24:28Z Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions Quesne, C. The D-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing D auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and w(2D)⊕s sp(4D,R) dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky-Winternitz system. The action of generators on wavefunctions is given in explicit form for D=2. 2011 Article Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions / C. Quesne // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 90 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C35; 81R05; 81R12 DOI:10.3842/SIGMA.2011.035 http://dspace.nbuv.gov.ua/handle/123456789/146804 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 D-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing D auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and w(2D)⊕s sp(4D,R) dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky-Winternitz system. The action of generators on wavefunctions is given in explicit form for D=2.
format	Article
author	Quesne, C.
spellingShingle	Quesne, C. Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Quesne, C.
author_sort	Quesne, C.
title	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
title_short	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
title_full	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
title_fullStr	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
title_full_unstemmed	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
title_sort	revisiting the symmetries of the quantum smorodinsky-winternitz system in d dimensions
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146804
citation_txt	Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions / C. Quesne // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 90 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT quesnec revisitingthesymmetriesofthequantumsmorodinskywinternitzsysteminddimensions
first_indexed	2025-07-11T00:38:06Z
last_indexed	2025-07-11T00:38:06Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 035, 21 pages Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System in D Dimensions? Christiane QUESNE Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium E-mail: cquesne@ulb.ac.be Received January 17, 2011, in final form March 25, 2011; Published online April 02, 2011 doi:10.3842/SIGMA.2011.035 Abstract. The D-dimensional Smorodinsky–Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introdu- cing D auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and w(2D) ⊕s sp(4D,R) dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky–Winternitz system. The action of generators on wavefunctions is given in explicit form for D = 2. Key words: Schrödinger equation; superintegrability; potential algebras; dynamical potential algebras 2010 Mathematics Subject Classification: 20C35; 81R05; 81R12 1 Introduction In classical mechanics, a Hamiltonian H with D degrees of freedom is said to be completely integrable if it allows D integrals of motion Xµ, µ = 1, 2, . . . , D, that are well-defined functions on phase space, are in involution and are functionally independent (see, e.g., [1]). These include the Hamiltonian, so that we may assume XD = H. The system is superintegrable if there exist k additional integrals of motion Yν , ν = 1, 2, . . . , k, 1 ≤ k ≤ D − 1, that are also well-defined functions on phase space and are such that the integrals H, X1, X2, . . . , XD−1, Y1, Y2, . . . , Yk are functionally independent. The cases k = 1 and k = D − 1 correspond to minimal and maximal superintegrability, respectively. Similar definitions apply in quantum mechanics with Poisson brackets replaced by commuta- tors, but H, Xµ, and Yν must now be well-defined operators forming an algebraically indepen- dent set. Maximally superintegrable quantum systems appear in many domains of physics, such as condensed matter as well as atomic, molecular, and nuclear physics. They have a lot of nice properties: they can be exactly (or quasi-exactly) solved, they are often separable in several co- ordinate systems and their spectrum presents some “accidental” degeneracies, i.e., degeneracies that do not follow from the geometrical symmetries of the problem. The most familiar examples of such systems are the Kepler–Coulomb [2, 3, 4] and the oscilla- tor [5, 6] ones. Other well-known instances are those resulting from the first systematic search for superintegrable Hamiltonians on E2 carried out by Smorodinsky, Winternitz, and collabora- tors [7, 8, 9] and from its continuation by Evans on E3 [10]. These studies were restricted to those cases where the integrals of motion are first- or second-order polynomials in the momenta. Later ?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:cquesne@ulb.ac.be http://dx.doi.org/10.3842/SIGMA.2011.035 http://www.emis.de/journals/SIGMA/S4.html 2 C. Quesne on, many efforts have been devoted to arriving at a complete classification of these so-called second-order superintegrable systems (see, e.g., [11, 12, 13, 14, 15, 16, 17]). Only recently, the pioneering work of Drach [18, 19] on two-dimensional Hamiltonian systems with third-order integrals of motion has been continued [20, 21]. Nowadays the search for D- dimensional superintegrable systems with higher-order integrals of motion has become a very active field of research (see, e.g., [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]). In the present paper, we plan to re-examine from an algebraic viewpoint one of the clas- sical examples of D-dimensional superintegrable quantum systems, namely the Smorodinsky– Winternitz (SW) one [7, 8, 9, 10, 33, 34], which may be defined in Cartesian coordinates as H(k) = D∑ µ=1 ( −∂2xµ + k2µ x2µ + ω2x2µ ) . (1.1) Here ω, k1, k2, . . . , kD are some constants, which we assume to be real and positive. Several distinct algebraic methods may be used in connection with superintegrable systems. One of them is based on the fact that the integrals of motion generate a nonlinear algebra closing at some order [35, 36, 37]. It has been shown, for instance, that for two-dimensional second-order superintegrable systems with nondegenerate potential and the corresponding three- dimensional conformally flat systems, one gets a quadratic algebra closing at order 6 [11, 12, 13, 14, 15]. Its finite-dimensional unitary representations can be determined [38] by using a deformed parafermion oscillator realization [39, 40], thereby allowing a calculation of the energy spectrum. This procedure can be extended to higher-order integrals of motion and to the corresponding higher-degree nonlinear algebras [25, 26]. Superintegrable systems may also be related [25, 26] to systems studied in supersymmetric quantum mechanics [41, 42] or higher-order supersymmetric quantum mechanics [43, 44, 45, 46, 47, 48, 49, 50, 51], hence can be described in terms of either linear or nonlinear superalgebras. As a consequence, supersymmetry provides a convenient tool for generating superintegrable quantum systems with higher-order integrals of motion [52, 53]. The concept of exact or quasi-exact solvability [54, 55, 56], based on the existence of an infinite flag of functionally linear spaces preserved by the Hamiltonian or only that of one of these spaces, appears to be related to finite-dimensional representations of some Lie algebras of first-order differential operators, such as sl(2,R), sl(3,R), etc. Although different from the concept of superintegrability, it can be related to the latter for some superintegrable systems (see, e.g., [27, 49, 57]). It is worth noting, however, that some alternative definitions of exact and quasi-exact solvability have been proposed for some specific superintegrable systems in connection with multiseparability of the corresponding Schrödinger equation [58, 59]. The accidental degeneracies appearing in the bound-state spectrum of superintegrable quan- tum systems may be understood in terms of a symmetry algebra, which is such that for any energy level the wavefunctions corresponding to degenerate states span the carrier space of one of its unitary irreducible representations (unirreps) [60, 61]. The generators of this symme- try algebra, commuting with the Hamiltonian, are integrals of motion, which may assume a rather complicated form in terms of some basic ones due to the fact that linear algebras are often preferred1 (note, however, that nonlinear algebras may also be considered [62]). A familiar example of this phenomenon is provided by the so(4) symmetry algebra of the three-dimensional Kepler–Coulomb problem [2, 3, 4]. Another one corresponds to the su(3) symmetry algebra of the three-dimensional SW system [34] (or, in general, su(D) for the D-dimensional one). In some cases, the symmetry algebra can be enlarged to a spectrum generating algebra (also called dynamical algebra) by including some ladder operators, which are not integrals of motion 1It is worth observing here that this may be seen as the obverse of the approach used in [11, 12, 13, 14, 15, 35, 36, 37, 38], where the generators are the basic integrals of motion but the algebra turns out to be nonlinear. Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 3 but act as raising or lowering operators on the bound-state wavefunctions in such a way that all of them carry a single unirrep of the algebra [63, 64, 65]. For the three-dimensional SW system, it has been shown [34] to be given by the semidirect sum Lie algebra w(3) ⊕s sp(6,R), where w(3) denotes a Weyl algebra (or, in general, by w(D)⊕s sp(2D,R) in D dimensions). For one-dimensional systems, three other types of Lie algebraic approaches have been ex- tensively studied. All of them rely on an embedding of the system into a higher-dimensional space by introducing some auxiliary continuous variables and on the subsequent reduction of the extended system to the initial one, a procedure also used in discussing superintegrability (see, e.g., [24]). They work for hierarchies of Hamiltonians, whose members correspond to the same potential but different quantized strengths. The simplest ones are the potential algebras [66, 67, 68], whose unirrep carrier spaces are spanned by wavefunctions with the same energy, but different potential strengths. Larger algebras, which also contain some generators connec- ting wavefunctions with different energies, are called dynamical potential algebras [69, 70, 71]2. Finally, a third kind of algebras, termed satellite algebras [74, 75], have the property that there is a conserved quantity different from the energy. Up to now, only the first one of these Lie algebraic approaches, namely that of potential algebras, has been applied to some D-dimensional superintegrable systems [76, 77, 78, 79, 80, 81, 82, 83]. The purpose of the present paper is threefold: first to apply this technique to the D- dimensional SW Hamiltonian (1.1), second to present for the same the first construction of a dynamical potential algebra in more than one dimension, and third to show very explicitly the action of both the potential and dynamical potential algebra generators on the wavefunctions in the two-dimensional case. The paper is organized as follows. In Section 2, the solutions, as well as the symmetry and dynamical algebras, of a 2D-dimensional harmonic oscillator are obtained in a suitable orthogonal coordinate system. In Section 3, they are transformed into the solutions, as well as the potential and dynamical potential algebras, of the D-dimensional SW system in hyperspherical coordinates. The D = 2 case is then dealt with in detail in Section 4. Finally, Section 5 contains the conclusion. 2 2D-dimensional harmonic oscillator Let us consider a harmonic oscillator Hamiltonian Hosc = 2D∑ µ=1 ( − ∂2Xµ +X2 µ ) in a 2D-dimensional space, whose Cartesian coordinates are denoted by Xµ, µ = 1, 2, . . . , 2D. For our purposes, it is convenient to consider it in a different orthogonal coordinate system, which we will now proceed to introduce. 2.1 Harmonic oscillator in variables R, θ1, θ2, . . . , θD−1, λ1, λ2, . . . , λD On making the change of variables X1 = R sin θ1 sin θ2 · · · sin θD−1 sinλ1, X2 = R sin θ1 sin θ2 · · · sin θD−1 cosλ1, 2Some authors prefer to use the terminology of dynamical algebra of the hierarchy instead of dynamical potential algebra and to employ discrete variables, related to the quantum numbers characterizing the system, instead of continuous auxiliary variables. In this way, they get discrete-differential realizations of the algebras [72]. Other authors favour the use of nonlinear superalgebras [73]. 4 C. Quesne X2ν−1 = R sin θ1 sin θ2 · · · sin θD−ν cos θD−ν+1 sinλν , ν = 2, 3, . . . , D − 1, X2ν = R sin θ1 sin θ2 · · · sin θD−ν cos θD−ν+1 cosλν , ν = 2, 3, . . . , D − 1, X2D−1 = R cos θ1 sinλD, X2D = R cos θ1 cosλD, (2.1) where 0 ≤ R <∞, 0 ≤ θν < π 2 , ν = 1, 2, . . . , D − 1, and 0 ≤ λν < 2π, ν = 1, 2, . . . , D, Hosc can be rewritten as Hosc = −∂2R − 2D − 1 R ∂R − 1 R2 { ∂2θ1 + [(2D − 3) cot θ1 − tan θ1]∂θ1 + D−1∑ ν=2 1 sin2 θ1 sin2 θ2 · · · sin2 θν−1 [ ∂2θν + [(2D − 2ν − 1) cot θν − tan θν ]∂θν ] + 1 sin2 θ1 sin2 θ2 · · · sin2 θD−1 ∂2λ1 + D−1∑ ν=2 1 sin2 θ1 sin2 θ2 · · · sin2 θD−ν cos2 θD−ν+1 ∂2λν + 1 cos2 θ1 ∂2λD } +R2 and is clearly separable. In the corresponding Schrödinger equation HoscΨosc(R,θ,λ) = EoscΨosc(R,θ,λ) (2.2) with θ = θ1θ2 · · · θD−1 and λ = λ1λ2 · · ·λD, we may therefore write Ψosc(R,θ,λ) = N oscL(z) ( D−1∏ ν=1 Θν(θν) )( D∏ ν=1 eipD−ν+1λν ) , z = R2, (2.3) where ∂2λνΨosc(R,θ,λ) = −p2D−ν+1Ψ osc(R,θ,λ) and p1, p2, . . . , pD ∈ Z. The normalization constant N osc in (2.3) will be determined in such a way that∫ dV \|Ψosc(R,θ,λ)\|2 = 1, (2.4) where dV = 2D∏ µ=1 dXµ = R2D−1dR [ D−1∏ ν=1 (sin θν)2D−2ν−1 cos θνdθν ]( D∏ ν=1 dλν ) . (2.5) As shown in the appendix, the angular part of wavefunctions (2.3) can be written as Θ (p) n (θ) = D−1∏ ν=1 Θ(aν ,bν) nν (θν), n = n1n2 · · ·nD−1, p = p1p2 · · · pD, (2.6) Θ(aν ,bν) nν (θν) = (cos θν)aν− 1 2 (sin θν)bν− 1 2P (aν− 1 2 ,bν+D−ν− 3 2) nν (− cos 2θν), (2.7) where n1, n2, . . . , nD−1 ∈ N, aν = \|pν \|+ 1 2 , ν = 1, 2, . . . , D − 1, Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 5 bν = 2nν+1+ 2nν+2+ · · ·+ 2nD−1+ \|pν+1\|+ \|pν+2\|+ · · ·+ \|pD\|+ 1 2 , ν = 1, 2, . . . , D − 2, bD−1 = \|pD\|+ 1 2 , (2.8) and P (aν− 1 2 ,bν+D−ν− 3 2) nν (− cos 2θν) denotes a Jacobi polynomial [84], while the radial part can be expressed as L(j)nr (z) = zjL(2j+D−1) nr (z)e− 1 2 z, (2.9) in terms of a Laguerre polynomial [84]. Here nr ∈ N, while j is defined by j = n1 + n2 + · · ·+ nD−1 + 1 2(\|p1\|+ \|p2\|+ · · ·+ \|pD\|) (2.10) and may take nonnegative integer or half-integer values. The corresponding energy eigenvalues are given by Eosc nrj = 2(2nr + 2j +D). (2.11) We therefore recover the well-known spectrum of the 2D-dimensional harmonic oscillator Eosc N = 2(N +D), N = 2nr + 2j = 0, 1, 2, . . . , whose levels, completely characterized by N , have a degeneracy equal to ( N+2D−1 2D−1 ) . Finally, the normalization constant in equation (2.3) can be easily determined from some well-known properties of Laguerre and Jacobi polynomials [84] and is given by N osc nrnp = ( nr! πD(nr + 2j +D − 1)! )1/2 × D−1∏ ν=1 ( nν !(2nν + aν + bν +D − ν − 1)(nν + aν + bν +D − ν − 2)!( nν + aν − 1 2 ) ! ( nν + bν +D − ν − 3 2 ) ! )1/2 . (2.12) 2.2 Harmonic oscillator symmetry and dynamical algebras As it is well known [5, 6], to each of the oscillator levels specified by N there corresponds a symmetric unirrep [N ] of its su(2D) symmetry algebra. The generators of the latter Ēµν = Eµν − 1 2D δµ,ν ∑ ρ Eρρ, µ, ν = 1, 2, . . . , 2D, with [ Ēµν , Ēµ′ν′ ] = δν,µ′Ēµν′ − δµ,ν′Ēµ′ν , Ē†µν = Ēνµ, are most easily constructed in terms of bosonic creation and annihilation operators α†µ = 1√ 2 ( Xµ − ∂Xµ ) , αµ = 1√ 2 ( Xµ + ∂Xµ ) , µ = 1, 2, . . . , 2D, (2.13) from Eµν = 1 2{α † µ, αν} = α†µαν + 1 2δµ,ν . (2.14) The harmonic oscillator Hamiltonian turns out to be proportional to the first-order Casimir operator C1 of u(2D), Hosc = 2C1 = 2 ∑ µ Eµµ = 2E . (2.15) 6 C. Quesne In the coordinates (2.1) chosen to describe the oscillator, the so(2D) subalgebra of su(2D), generated by Lµν = −i ( Ēµν − Ēνµ ) = −i(Eµν − Eνµ), (2.16) such that [Lµν , Lµ′ν′ ] = i(δµ,µ′Lνν′ − δµ,ν′Lνµ′ − δν,µ′Lµν′ + δν,ν′Lµµ′), L†µν = Lµν = −Lνµ, is explicitly reduced. Its unirreps are characterized by 2j, which runs over N,N −2, . . . , 0 (or 1) for a given N . The remaining generators of su(2D) may be taken as Tµν = Ēµν + Ēνµ. (2.17) The operators D†µν = α†µα † ν , Dµν = αµαν (2.18) act as raising and lowering operators relating among themselves wavefunctions corresponding to even or odd values of N . Together with Eµν , they generate an sp(4D,R) Lie algebra, whose (nonvanishing) commutation relations are given by [Eµν , Eµ′ν′ ] = δν,µ′Eµν′ − δµ,ν′Eµ′ν , [Eµν , D † µ′ν′ ] = δν,µ′D † µν′ + δν,ν′D † µµ′ , [Eµν , Dµ′ν′ ] = −δµ,µ′Dνν′ − δµ,ν′Dνµ′ , [Dµν , D † µ′ν′ ] = δµ,µ′Eν′ν + δµ,ν′Eµ′ν + δν,µ′Eν′µ + δν,ν′Eµ′µ. To connect the wavefunctions with an even N value to those with an odd one, we have to use the bosonic creation and annihilation operators (2.13), which generate a Weyl algebra w(2D), specified by [αµ, α † ν ] = δµ,νI. The whole set of operators {Eµν , D†µν , Dµν , α † µ, αµ, I} then provides us with the harmonic oscil- lator dynamical algebra, which is the semidirect sum Lie algebra w(2D)⊕s sp(4D,R), as shown by the remaining (nonvanishing) commutation relations [Eµν , α † µ′ ] = δν,µ′α † µ, [Eµν , αµ′ ] = −δµ,µ′αν , [Dµν , α † µ′ ] = δµ,µ′αν + δν,µ′αµ, [D†µν , αµ′ ] = −δµ,µ′α†ν − δν,µ′α†µ. To apply the symmetry and dynamical algebra generators to the oscillator wavefunctions (2.3) written in the variables R, θ, λ, we have to express the creation and annihilation operators α†µ, αµ in such variables. This implies combining the transformation (2.1) with the corresponding change for the partial differential operators ∂X2ν−1 = sinλν∂ (ν,1) + cosλν∂ (ν,2), ∂X2ν = cosλν∂ (ν,1) − sinλν∂ (ν,2), ν = 1, 2, . . . , D, (2.19) where ∂(1,1) = sin θ1 sin θ2 · · · sin θD−1∂R Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 7 + 1 R D−1∑ ρ=1 csc θ1 csc θ2 · · · csc θρ−1 cos θρ sin θρ+1 sin θρ+2 · · · sin θD−1∂θρ , ∂(ν,1) = sin θ1 sin θ2 · · · sin θD−ν cos θD−ν+1∂R + 1 R D−ν∑ ρ=1 csc θ1 csc θ2 · · · csc θρ−1 cos θρ sin θρ+1 sin θρ+2 · · · sin θD−ν cos θD−ν+1∂θρ − 1 R csc θ1 csc θ2 · · · csc θD−ν sin θD−ν+1∂θD−ν+1 , ν = 2, 3, . . . , D − 1, ∂(D,1) = cos θ1∂R − 1 R sin θ1∂θ1 , and ∂(1,2) = 1 R csc θ1 csc θ2 · · · csc θD−1∂λ1 , ∂(ν,2) = 1 R csc θ1 csc θ2 · · · csc θD−ν sec θD−ν+1∂λν , ν = 2, 3, . . . , D − 1, ∂(D,2) = 1 R sec θ1∂λD . We shall carry out this transformation explicitly for D = 2 in Section 4. 3 Reduction of the 2D-dimensional harmonic oscillator to the D-dimensional SW system To go from the 2D-dimensional harmonic oscillator Hamiltonian Hosc to some extended SW Hamiltonian H, let us first transform the original Cartesian coordinates Xµ, µ = 1, 2, . . . , 2D, into some new ones xµ, µ = 1, 2, . . . , 2D, such that x1 = r sinφ1 sinφ2 · · · sinφD−1, xν = r sinφ1 sinφ2 · · · sinφD−ν cosφD−ν+1, ν = 2, 3, . . . , D − 1, xD = r cosφ1, xD+ν = λν , ν = 1, 2, . . . , D, and R = √ ω r, θν = φν , ν = 1, 2, . . . , D − 1, 0 ≤ r <∞, 0 ≤ φν < π 2 , ν = 1, 2, . . . , D − 1, 0 ≤ λν < 2π, ν = 1, 2, . . . , D. Here r, φ1, φ2, . . . , φD−1 are hyperspherical coordinates in the D-dimensional subspace (x1, x2, . . . , xD). The volume element in the transformed 2D-dimensional space is given by dv = 2D∏ µ=1 dxµ = rD−1dr [ D−1∏ ν=1 (sinφν)D−ν−1dφν ]( D∏ ν=1 dλν ) . (3.1) On making next the change of function Ψ(r,φ,λ) = O1/2Ψosc(R,θ,λ), (3.2) with O = (ωr)D D−1∏ ν=1 (sinφν)D−ν cosφν , (3.3) 8 C. Quesne the harmonic oscillator wavefunctions Ψosc(R,θ,λ), living in a Hilbert space with measure dV given in (2.5), are mapped onto some functions Ψ(r,φ,λ), living in a Hilbert space with mea- sure dv defined in (3.1). As a consequence of (2.4), we obtain∫ dv \|Ψ(r,φ,λ)\|2 = 1. By this unitary transformation, the harmonic oscillator Hamiltonian Hosc is changed into H/ω = O1/2HoscO−1/2 (3.4) and similarly for other operators acting in the harmonic oscillator Hilbert space. A straightfor- ward calculation leads to the result H = −∂2r − D − 1 r ∂r − 1 r2 { ∂2φ1 + (D − 2) cotφ1∂φ1 + D−1∑ ν=2 1 sin2 φ1 sin2 φ2 · · · sin2 φν−1 [ ∂2φν + (D − ν − 1) cotφν∂φν ] + 1 sin2 φ1 sin2 φ2 · · · sin2 φD−1 ( ∂2λ1 + 1 4 ) + D−1∑ ν=2 1 sin2 φ1 sin2 φ2 · · · sin2 φD−ν cos2 φD−ν+1 ( ∂2λν + 1 4 ) + 1 cos2 φ1 ( ∂2λD + 1 4 )} + ω2r2. (3.5) The eigenvalues of H are directly obtained from (2.11) as Enrj = 2ω(2nr + 2j +D), nr = 0, 1, 2, . . . , j = 0, 12 , 1, 3 2 , . . . . (3.6) The corresponding wavefunctions can be derived from (2.3), (2.6), (2.7), (2.9), (2.12), (3.2), and (3.3) and read Ψnrnp(r,φ,λ) = NnrnpZ(j) nr (z)Φ (p) n (φ) ( D∏ ν=1 eipD−ν+1λν ) , Z(j) nr (z) = ( z ω )j+D 4 L(2j+D−1) nr (z)e− 1 2 z, z = ωr2, Φ (p) n (φ) = D−1∏ ν=1 Φ(aν ,bν) nν (φν) = D−1∏ ν=1 (cosφν)aν (sinφν)bν+ 1 2 (D−ν−1)P (aν− 1 2 ,bν+D−ν− 3 2) nν (− cos 2φν), Nnrnp = ωj+ D 2 N osc nrnp, (3.7) where n = n1n2 · · ·nD−1, p = p1p2 · · · pD, nr, n1, n2, . . . , nD−1 ∈ N, p1, p2, . . . , pD ∈ Z, while j, and aν , bν are defined in (2.10) and (2.8), respectively. In the subspace of functions Ψnrnp(r,φ,λ) with fixed p, the Hamiltonian H, defined in (3.5), has the same action as the D-dimensional Hamiltonian H(k) = −∂2r − D − 1 r ∂r − 1 r2 { ∂2φ1 + (D − 2) cotφ1∂φ1 Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 9 + D−1∑ ν=2 1 sin2 φ1 sin2 φ2 · · · sin2 φν−1 [ ∂2φν + (D − ν − 1) cotφν∂φν ]} + k21 r2 sin2 φ1 sin2 φ2 · · · sin2 φD−1 + D−1∑ ν=2 k2ν r2 sin2 φ1 sin2 φ2 · · · sin2 φD−ν cos2 φD−ν+1 + k2D r2 cos2 φ1 + ω2r2, where we have defined k = k1k2 · · · kD and kν = √ p2D−ν+1 − 1 4 , ν = 1, 2, . . . , D. (3.8) The latter Hamiltonian is but the SW one (1.1), expressed in hyperspherical coordinates r, φ1, φ2, . . . , φD−1. We conclude that H is an extension of H(k), resulting from the introduction of D auxiliary continuous variables λν = xD+ν , ν = 1, 2, . . . , D, and that, conversely, H(k) is obtained from H by projecting it down into the D-dimensional subspace (x1, x2, . . . , xD).3 As a by-product of this reduction process, we have determined the wavefunctions Ψ (k) nrn(r,φ) of H(k) in hyperspherical coordinates. Equation (3.7) may indeed be rewritten as Ψnrnp(r,φ,λ) = Ψ (k) nrn(r,φ)(2π)−D/2 D∏ ν=1 eipD−ν+1λν , Ψ (k) nrn(r,φ) = N (k) nrnZ(j) nr (z)Φ (p) n (φ), N (k) nrn = (2π)D/2Nnrnp, (3.9) with k and p related as in (3.8). By a transformation similar to (3.4), the generators Ēµν of the harmonic oscillator symmetry algebra su(2D) are changed into some operators acting on Ψnrnp(r,φ,λ). Since the latter may change nr and j separately (provided their sum nr + j = N/2 is preserved), this means in particular (see equation (2.10)) that the pν ’s (hence the kν ’s) may change too. The transformed su(2D) algebra may therefore connect among themselves some wavefunctions of H belonging to the same energy eigenvalue (3.6), but associated with different reduced Hamiltonians H(k). We conclude that it provides us with a potential algebra for the SW system. Similarly, the transformed w(2D)⊕s sp(4D,R) algebra will be a dynamical potential algebra for the same. As a final point, it is worth observing that in the harmonic oscillator wavefunctions (2.3), the quantum numbers pν , ν = 1, 2, . . . , D, run over Z. On the other hand, in (1.1), the parameters kν , ν = 1, 2, . . . , D, have been assumed real and positive. From equation (3.8), however, it is clear that pD−ν+1 = 0 would lead to an imaginary value of kν and to unphysical wavefunctions (3.9), while \|pD−ν+1\| and −\|pD−ν+1\| with \|pD−ν+1\| ≥ 1 would give rise to the same kν , hence to some replicas of physical wavefunctions (3.9). The correspondence between the harmonic oscillator wavefunctions and the extended SW ones is therefore not one-to-one. This lack of bijectiveness is a known aspect of potential algebraic approaches (see [69] where this phenomenon was first pointed out). 4 The two-dimensional case 4.1 Harmonic oscillator symmetry and dynamical algebras To deal in detail with the two-dimensional case, it is appropriate to rewrite the four-dimensional harmonic oscillator wavefunctions Ψosc nr,n,p1,p2(R, θ, λ1, λ2) (with j = n + 1 2(\|p1\| + \|p2\|)) in an 3Strictly speaking, this is true only for those kν ’s that can be written in the form (3.8) with integer p2D−ν+1. 10 C. Quesne equivalent form Ψ̄osc nr,j,m,m′ (R, θ, λ1, λ2) using either hyperspherical harmonics Y2j,m,m′(α, β, γ) or (complex conjugate) rotation matrix elements Dj∗ m,−m′(α, β, γ) expressed in terms of Euler angles α, β, γ [85], Y2j,m,m′(α, β, γ) = (−1)j−m ′ ( 2j + 1 2π2 )1/2 Dj∗ m,−m′(α, β, γ), Dj∗ m,−m′(α, β, γ) = eimαdjm,−m′(β)e−im ′γ . Here j runs over 0, 12 , 1, 3 2 , . . ., while m and m′ take values in the set {j, j − 1, . . . ,−j}. On setting θ = 1 2β, λ1 = 1 2(γ − α), λ2 = 1 2(γ + α), p1 = m−m′, p2 = −m−m′, or, conversely, α = λ2 − λ1, β = 2θ, γ = λ2 + λ1, m = 1 2(p1 − p2), m′ = −1 2(p1 + p2), and on using the relation between rotation functions djm,−m′(β) and Jacobi polynomials [85], we indeed get Ψosc nr,n,p1,p2(R, θ, λ1, λ2) = (−1) 1 2 (\|p1\|+p1)+\|p2\|Ψ̄osc nr,j,m,m′(R, θ, λ1, λ2) (4.1) with Ψ̄osc nr,j,m,m′(R, θ, λ1, λ2) = (−1)j−m ′ ( (2j + 1)nr! π2(nr + 2j + 1)! )1/2 L(j)nr (z) × djm,−m′(2θ)e −i(m+m′)λ1ei(m−m ′)λ2 , L(j)nr (z) = zjL(2j+1) nr (z)e− 1 2 z, z = R2. (4.2) The advantage of this new form is that the wavefunctions Ψosc nr,n,p1,p2(R, θ, λ1, λ2), which were classified according to su(4) ⊃ so(4) [N ] (2j) with N = 2nr + 2j and 2j = 2n+ \|p1\|+ \|p2\|, now turn out to be explicitly reduced with respect to su(4) ⊃ so(4) ' su(2)⊕ su(2) ⊃ u(1)⊕ u(1) [N ] (2j) ' [j]⊕ [j] [m]⊕ [m′] . (4.3) This will allow us to use the full machinery of angular momentum theory for determining the explicit action of the symmetry and dynamical algebra generators on wavefunctions. The two su(2) algebras appearing in chain (4.3) are generated by Ji and Ki, i = 1, 2, 3, defined in terms of Lµν , µ, ν = 1, 2, 3, 4, (see equation (2.16)) by Ji = 1 2 ( 1 2εijkLjk − Li4 ) , Ki = 1 2 ( 1 2εijkLjk + Li4 ) , (4.4) where i, j, k run over 1, 2, 3 and εijk is the antisymmetric tensor. The operators Ji and Ki satisfy the relations [Ji, Jj ] = iεijkJk, [Ki,Kj ] = iεijkKk, [Ji,Kj ] = 0, J†i = Ji, K†i = Ki. Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 11 Instead of the Cartesian components of J and K, we may use alternatively J0 = J3, J± = J1±iJ2, K0 = K3, K± = K1 ± iK2, with J0 and K0 generating the two u(1) subalgebras in (4.3). The differential operator form of J0, J±, K0, and K± can be obtained by combining equations (2.1), (2.13), (2.14), (2.16), (2.19), and (4.4) and is given by J0 = i 2(∂λ1 − ∂λ2), J± = 1 2e ∓i(λ1−λ2)[±∂θ − i(cot θ∂λ1 + tan θ∂λ2)], K0 = i 2(∂λ1 + ∂λ2), K± = 1 2e ∓i(λ1+λ2)[∓∂θ + i(cot θ∂λ1 − tan θ∂λ2)]. From some differential equation relations satisfied by rotation functions djm,−m′(2θ) [86], it is then easy to check that J0Ψ̄ osc nr,j,m,m′ = mΨ̄osc nr,j,m,m′ , J±Ψ̄osc nr,j,m,m′ = [(j ∓m)(j ±m+ 1)]1/2Ψ̄osc nr,j,m±1,m′ , K0Ψ̄ osc nr,j,m,m′ = m′Ψ̄osc nr,j,m,m′ , K±Ψ̄osc nr,j,m,m′ = [(j ∓m′)(j ±m′ + 1)]1/2Ψ̄osc nr,j,m,m′±1, which proves the above-mentioned result. It is now convenient to rewrite all operators of physical interest as components T (s,t) σ,τ , σ = s, s−1, . . . ,−s, τ = t, t−1, . . . ,−t, of irreducible tensors of rank (s, t) with respect to su(2)⊕su(2). These must satisfy commutation relations of the type[ J0, T (s,t) σ,τ ] = σT (s,t) σ,τ , [ J±, T (s,t) σ,τ ] = [(s∓ σ)(s± σ + 1)]1/2T (s,t) σ±1,τ ,[ K0, T (s,t) σ,τ ] = τT (s,t) σ,τ , [ K±, T (s,t) σ,τ ] = [(t∓ τ)(t± τ + 1)]1/2T (s,t) σ,τ±1. Since the bosonic creation and annihilation operators serve as building blocks for the con- struction of other operators, let us start with them. The creation operators can be written as components A†σ,τ , σ, τ = 1 2 , −1 2 , of an irreducible tensor of rank ( 1 2 , 1 2 ) , A†± 1 2 ,± 1 2 = ∓ 1√ 2 ( α†1 ± iα†2 ) , A†± 1 2 ,∓ 1 2 = 1√ 2 ( α†3 ∓ iα†4 ) . (4.5) The same is true for the annihilation operators, the corresponding components being given by Aσ,τ = (−1)1−σ−τ ( A†−σ,−τ )† , σ, τ = 1 2 ,− 1 2 . (4.6) On coupling an operator A† with an operator A according to[ A† ×A ]s,t σ,τ = ∑ σ′,τ ′ 〈 1 2 σ ′, 12 σ − σ ′∣∣s σ〉〈12 τ ′, 12 τ − τ ′∣∣t τ〉A†σ′,τ ′Aσ−σ′,τ−τ ′ , where 〈 , \| 〉 denotes an su(2) Wigner coefficient [85], we obtain the su(4) symmetry algebra generators classified with respect to chain (4.3). These include Jσ = [ A† ×A ]1,0 σ,0 , Kτ = [ A† ×A ]0,1 0,τ , σ, τ = +1, 0,−1, with J±1 = ∓J±/ √ 2 and K±1 = ∓K±/ √ 2, as well as the nine components of an irreducible tensor of rank (1, 1), Tσ,τ = [ A† ×A ]1,1 σ,τ , σ, τ = +1, 0,−1. (4.7) The latter may be written as T±1,±1 = −1 4(T11 ± 2iT12 − T22), T±1,0 = 1 2 √ 2 (±T13 − iT14 + iT23 ± T24), T±1,∓1 = −1 4(T33 ∓ 2iT34 − T44), T0,±1 = 1 2 √ 2 (±T13 + iT14 + iT23 ∓ T24), 12 C. Quesne T0,0 = 1 2(T11 + T22) = −1 2(T33 + T44) in terms of the operators Tµν , defined in (2.17). Observe that the u(4) first-order Casimir operator (2.15) is, up to some constants, the scalar that can be obtained in such a coupling procedure,[ A† ×A ]0,0 0,0 = 1 2(E − 2). Similarly, the coupling of two operators A† provides us with the raising operators belonging to sp(8,R), D† = A† · A† = −2 [ A† ×A† ]0,0 0,0 , D†σ,τ = [ A† ×A† ]1,1 σ,τ , σ, τ = +1, 0,−1, or, in detail, D† = D†11 +D†22 +D†33 +D†44 and D†±1,±1 = 1 2 ( D†11 ± 2iD†12 −D † 22 ) , D†±1,0 = − 1√ 2 ( ±D†13 − iD†14 + iD†23 ±D † 24 ) , D†±1,∓1 = 1 2 ( D†33 ∓ 2iD†34 −D † 44 ) , D†0,±1 = − 1√ 2 ( ±D†13 + iD†14 + iD†23 ∓D † 24 ) , D†0,0 = 1 2 ( −D†11 −D † 22 +D†33 +D†44 ) in terms of D†µν defined in (2.18). The corresponding lowering operators are then D = ( D† )† , Dσ,τ = (−1)σ+τ ( D†−σ,−τ )† , σ, τ = +1, 0,−1. It is now straightforward to determine the action of A†σ,τ on the wavefunctions Ψ̄osc nr,j,m,m′ (R, θ, λ1, λ2). Application of the Wigner–Eckart theorem with respect to su(2)⊕su(2) [85] indeed leads to the relation A†σ,τ Ψ̄osc nr,j,m,m′ = ∑ n′r,j ′ 〈 n′r, j ′∥∥A†∥∥nr, j〉〈j m, 12 σ∣∣j′m+ σ 〉〈 j m′, 12 τ ∣∣j′m′ + τ 〉 × Ψ̄osc n′r,j ′,m+σ,m′+τ , (4.8) where 〈 n′r, j ′∥∥A†∥∥nr, j〉 denotes a reduced matrix element, the summation over j′ runs over j + 1 2 , j − 1 2 , and n′r is determined by the selection rule n′r + j′ = nr + j + 1 2 implying that n′r = nr, nr + 1, respectively. To calculate the two independent reduced matrix elements, it is enough to consider equation (4.8) for the special case m = m′ = j and to use the differential operator form of A†± 1 2 ,± 1 2 , A†± 1 2 ,± 1 2 = 1 2 e∓iλ1 [ i ( sin θ∂R + 1 R cos θ∂θ ) ± 1 R csc θ∂λ1 − iR sin θ ] , following from (2.1), (2.13), (2.19), and (4.5). Simple properties of the rotation function djm,−m′(2θ) and of the Laguerre polynomial L (2j+1) nr (z) then lead to the results 〈 nr, j + 1 2 ∥∥A†∥∥nr, j〉 = i ( (2j + 1)(nr + 2j + 2) 2j + 2 )1/2 , Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 13 〈 nr + 1, j − 1 2 ∥∥A†∥∥nr, j〉 = −i ( (2j + 1)(nr + 1) 2j )1/2 . (4.9) The operators Aσ,τ satisfy an equation similar to (4.8) with 〈 n′r, j ′∥∥A†∥∥nr, j〉 replaced by〈 n′r, j ′∥∥A∥∥nr, j〉 and n′r = nr − 1, nr for j′ = j + 1 2 , j − 1 2 , respectively. The corresponding reduced matrix elements can be directly calculated from the relation 〈 n′r, j ′∥∥A∥∥nr, j〉 = 2j + 1 2j′ + 1 〈 nr, j ∥∥A†∥∥n′r, j′〉∗, (4.10) which is a direct consequence of (4.6). For the su(4) generators that do not belong to so(4), we get the equation Tσ,τ Ψ̄osc nr,j,m,m′ = ∑ n′r,j ′ 〈 n′r, j ′∥∥T ∥∥nr, j〉〈j m, 1σ∣∣j′m+ σ 〉〈 j m′, 1 τ ∣∣j′m′ + τ 〉 × Ψ̄osc n′r,j ′,m+σ,m′+τ , (4.11) where j′ = j+1, j, j−1 and n′r = nr−1, nr, nr+1, respectively. Equation (4.7) and the coupling law for reduced matrix elements [85] enable us to determine 〈 nr − 1, j + 1 ∥∥T ∥∥nr, j〉 = − ( (2j + 1)nr(nr + 2j + 2) 2j + 3 )1/2 ,〈 nr, j ∥∥T ∥∥nr, j〉 = nr + j + 1,〈 nr + 1, j − 1 ∥∥T ∥∥nr, j〉 = − ( (2j + 1)(nr + 1)(nr + 2j + 1 2j − 1 )1/2 from (4.9) and (4.10). The operators D†σ,τ and Dσ,τ satisfy a relation similar to (4.11) with 〈 nr, j + 1 ∥∥D†∥∥nr, j〉 = − ( (2j + 1)(nr + 2j + 2)(nr + 2j + 3) 2j + 3 )1/2 ,〈 nr + 1, j ∥∥D†∥∥nr, j〉 = [(nr + 1)(nr + 2j + 2)]1/2,〈 nr + 2, j − 1 ∥∥D†∥∥nr, j〉 = − ( (2j + 1)(nr + 1)(nr + 2) 2j − 1 )1/2 , and 〈 n′r, j ′∥∥D∥∥nr, j〉 obtained from these as in (4.10). Finally, with the equations D†Ψ̄osc nr,j,m,m′ = −2[(nr + 1)(nr + 2j + 2)]1/2Ψ̄osc nr+1,j,m,m′ , DΨ̄osc nr,j,m,m′ = −2[nr(nr + 2j + 1)]1/2Ψ̄osc nr−1,j,m,m′ , the action of the harmonic oscillator symmetry and dynamical algebra generators on Ψ̄osc nr,j,m,m′ (R, θ, λ1, λ2) is completely determined. 4.2 SW system potential and dynamical potential algebras In two dimensions, equations (2.8) and (3.8) simply lead to k21 = b(b− 1), k22 = a(a− 1). 14 C. Quesne In the following, it will prove convenient to use a and b instead of k1 and k2. Up to the same phase factor as that occurring in (4.1), the extended SW Hamiltonian wavefunctions (3.9) can then be rewritten as4 Ψ̄nr,n,a,b(r, φ, λ1, λ2) = Ψ̄(a,b) nr,n(r, φ)(2π)−1ei(b− 1 2)λ1ei(a− 1 2)λ2 , Ψ̄(a,b) nr,n(r, φ) = N (a,b) nr,n Z (j) nr (z)Φ(a,b) n (φ), Z(j) nr (z) = ( z ω )n+ 1 2 (a+b) L(2n+a+b) nr (z)e− 1 2 z, Φ(a,b) n (φ) = cosa φ sinb φP (a− 1 2 ,b− 1 2) n (− cos 2φ), N (a,b) nr,n = (−1)a+b−12 ( ω2n+a+b+1nr!n! (2n+ a+ b)(n+ a+ b− 1)! (nr + 2n+ a+ b)! ( n+ a− 1 2 ) ! ( n+ b− 1 2 ) ! )1/2 , where n, a, b are related to j, m, m′ used in (4.2) through the relations5 j = n+ 1 2(a+ b− 1), m = 1 2(a− b), m′ = −1 2(a+ b− 1), (4.12) or, conversely, a = m−m′ + 1 2 , b = −m−m′ + 1 2 , n = j +m′. The generators of the su(4) potential algebra, as well as those of the w(4)⊕ssp(8,R) dynamical potential algebra, can be directly obtained by performing transformation (3.4) on the operators of Section 4.1. We get for instance6 J0 = i 2 (∂λ1 − ∂λ2), K0 = i 2 (∂λ1 + ∂λ2), J± = 1 2 e∓i(λ1−λ2) [ ±∂φ − cotφ ( i∂λ1 ± 1 2 ) − tanφ ( i∂λ2 ∓ 1 2 )] , K± = 1 2 e∓i(λ1+λ2) [ ∓∂φ + cotφ ( i∂λ1 ± 1 2 ) − tanφ ( i∂λ2 ± 1 2 )] , T+1,+1 = 1 4ω e−2iλ1 [ − sin2 φ∂2r − 2 r sinφ cosφ∂2rφ + 2i r ∂2rλ1 − 1 r2 cos2 φ∂2φ + 2i r2 cotφ∂2φλ1 + 1 r2 csc2 φ∂2λ1 + 1 r (1 + sin2 φ)∂r + 2 r2 (cotφ+ sinφ cosφ)∂φ − 3i r2 csc2 φ∂λ1 − 5 4r2 csc2 φ+ ω2r2 sin2 φ ] , A†± 1 2 ,± 1 2 = 1 2 e∓iλ1 [ i ( sinφ∂r + 1 r cosφ∂φ ) ± 1 r cscφ∂λ1 − i 2r cscφ− ir sinφ ] , D†+1,+1 = 1 4ω e−2iλ1 { − sin2 φ∂2r − 2 r sinφ cosφ∂2rφ + 2i r ∂2rλ1 − 1 r2 cos2 φ∂2φ + 2i r2 cotφ∂2φλ1 + 1 r2 csc2 φ∂2λ1 + [ 1 r (1 + sin2 φ) + 2ωr sin2 φ ] ∂r + 2 [ 1 r2 (cotφ+ sinφ cosφ) + ω sinφ cosφ ] ∂φ − i ( 3 r2 csc2 φ+ 2ω ) ∂λ1 − 5 4r2 csc2 φ− ω2r2 sin2 φ− ω } , 4It is worth observing here that integer or half-integer values of j, m, and m′ are related to integer values of n and half-integer ones of a and b. The results for matrix elements of potential and dynamical potential algebra generators are only valid for such a and b (see footnote 3), although those for wavefunctions are not restricted to these values provided factorials are replaced by gamma functions. 5Equation (4.12) is valid for positive p1 and p2, corresponding to physical wavefunctions (see discussion at the end of Section 3). 6For simplicity’s sake, we denote both types of operators by the same symbols. Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 15 D† = 1 2ω ( −H − 2ωr∂r + 2ω2r2 − 2ω ) . It is also straightforward to derive their matrix elements from the results of Section 4.1 and equation (4.12). We list them below: J0Ψ̄nr,n,a,b = 1 2(a− b)Ψ̄nr,n,a,b, K0Ψ̄nr,n,a,b = −1 2(a+ b− 1)Ψ̄nr,n,a,b, J+Ψ̄nr,n,a,b = [( n+ a+ 1 2 ) ( n+ b− 1 2 )]1/2 Ψ̄nr,n,a+1,b−1, J−Ψ̄nr,n,a,b = [( n+ a− 1 2 ) ( n+ b+ 1 2 )]1/2 Ψ̄nr,n,a−1,b+1, K+Ψ̄nr,n,a,b = [(n+ 1)(n+ a+ b− 1)]1/2Ψ̄nr,n+1,a−1,b−1, K−Ψ̄nr,n,a,b = [n(n+ a+ b)]1/2Ψ̄nr,n−1,a+1,b+1, Tσ,τ Ψ̄nr,n,a,b = n+τ+1∑ n′=n+τ−1 tn′(nr, 2n+ a+ b) × 〈 n+ 1 2(a+ b− 1) 1 2(a− b), 1 σ ∣∣n′ − τ + 1 2(a+ b− 1) 1 2(a− b) + σ 〉 × 〈 n+ 1 2(a+ b− 1) − 1 2(a+ b− 1), 1 τ ∣∣n′ − τ + 1 2(a+ b− 1) − 1 2(a+ b− 1) + τ 〉 × Ψ̄nr−(n′−n−τ),n′,a+σ−τ,b−σ−τ , A†σ,τ Ψ̄nr,n,a,b = n+τ+ 1 2∑ n′=n+τ− 1 2 an′(nr, 2n+ a+ b) × 〈 n+ 1 2(a+ b− 1) 1 2(a− b), 12 σ ∣∣n′ − τ + 1 2(a+ b− 1) 1 2(a− b) + σ 〉 × 〈 n+ 1 2(a+ b− 1) − 1 2(a+ b− 1), 12 τ ∣∣n′ − τ + 1 2(a+ b− 1) − 1 2(a+ b− 1) + τ 〉 × Ψ̄nr−(n′−n−τ)+ 1 2 ,n′,a+σ−τ,b−σ−τ , D†σ,τ Ψ̄nr,n,a,b = n+τ+1∑ n′=n+τ−1 dn′(nr, 2n+ a+ b) × 〈 n+ 1 2(a+ b− 1) 1 2(a− b), 1 σ ∣∣n′ − τ + 1 2(a+ b− 1) 1 2(a− b) + σ 〉 × 〈 n+ 1 2(a+ b− 1) − 1 2(a+ b− 1), 1 τ ∣∣n′ − τ + 1 2(a+ b− 1) − 1 2(a+ b− 1) + τ 〉 × Ψ̄nr−(n′−n−τ)+1,n′,a+σ−τ,b−σ−τ , D†Ψ̄nr,n,a,b = −2[(nr + 1)(nr + 2n+ a+ b+ 1)]1/2Ψ̄nr+1,n,a,b. Here tn′(nr, 2n+ a+ b) =  − ( (2n+a+b)nr(nr+2n+a+b+1) 2n+a+b+2 )1/2 if n′ = n+ τ + 1, nr + n+ 1 2(a+ b+ 1) if n′ = n+ τ , − ( (2n+a+b)(nr+1)(nr+2n+a+b) 2n+a+b−2 )1/2 if n′ = n+ τ − 1, an′(nr, 2n+ a+ b) = i ( (2n+a+b)(nr+2n+a+b+1) 2n+a+b+1 )1/2 if n′ = n+ τ + 1 2 , −i ( (2n+a+b)(nr+1) 2n+a+b−1 )1/2 if n′ = n+ τ − 1 2 , and dn′(nr, 2n+ a+ b) =  − ( (2n+a+b)(nr+2n+a+b+1)(nr+2n+a+b+2) 2n+a+b+2 )1/2 if n′ = n+ τ + 1, [(nr + 1)(nr + 2n+ a+ b+ 1)]1/2 if n′ = n+ τ , − ( (2n+a+b)(nr+1)(nr+2) 2n+a+b−2 )1/2 if n′ = n+ τ − 1. 16 C. Quesne From these results, we conclude that the potential algebra generators produce transitions between levels belonging to spectra of Hamiltonians characterized by parameters (a, b), (a± 1, b∓ 1), (a± 1, b± 1), (a± 2, b), and (a, b± 2). For the dynamical potential algebra generators, the same Hamiltonians are involved together with those associated with (a± 1, b) and (a, b± 1). 5 Conclusion In the present paper, we have re-examined the D-dimensional SW system, which may be con- sidered as the archetype of D-dimensional superintegrable system. We have completed Evans previous algebraic study, wherein its symmetry and dynamical algebras had been determined, by constructing its potential and dynamical potential algebras. In our approach based on the use of hyperspherical coordinates in the D-dimensional space and on the introduction of D auxiliary continuous variables, the SW system has been obtained by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and w(2D)⊕s sp(4D,R) dynamical algebras of the latter have then been transformed into correspon- ding potential and dynamical potential algebras for the former. Finally, the two-dimensional case has been studied in the fullest detail. Possible connections with other approaches currently used in connection with the SW system or, more generally, superintegrable systems, such as supersymmetry [52, 53], path integrals [87], coherent states [88], and deformations [89, 90], might be interesting topics for future investiga- tion. A Wavefunctions of the 2D-dimensional harmonic oscillator The purpose of this appendix is to derive the explicit form of the harmonic oscillator wavefunc- tions (2.3). On inserting (2.3) in the Schrödinger equation (2.2), the latter separates into D − 1 angular equations{ −d2θν − [(2D − 2ν − 1) cot θν − tan θν ]dθν + Cν+1 sin2 θν + p2ν cos2 θν − Cν } Θν(θν) = 0, ν = 1, 2, . . . , D − 1, (A.1) and a radial equation( −d2R − 2D − 1 R dR + C1 R2 +R2 − Eosc ) L(z) = 0. (A.2) Here C1, C2, . . . , CD−1 are D − 1 separation constants, while CD is defined by CD = p2D. (A.3) In the following, we are going to show that there does exist a solution to the whole set of D equations (A.1) and (A.2) such that all the separation constants Cν , ν = 1, 2, . . . , D − 1, are nonnegative. Let us start by solving the angular equation (A.1) corresponding to the variable θν in terms of pν and Cν+1. The ansatz Θν(θν) = (cos θν)aν− 1 2 (sin θν)bν− 1 2Fν(uν), uν = cos2 θν , transforms it into the hypergeometric differential equation [84]{ uν(1− uν)d2uν + [γ − (α+ β + 1)uν ]duν − αβ } Fν(uν) = 0 (A.4) Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System 17 provided we choose the constants aν and bν in such a way that( aν − 1 2 )2 = p2ν , ( bν − 1 2 ) ( bν + 2D − 2ν − 5 2 ) = Cν+1. (A.5) In (A.4), α, β, and γ are given by α = 1 2(aν + bν +D − ν − 1 + ∆ν), β = 1 2(aν + bν +D − ν − 1−∆ν), γ = aν + 1 2 , where ∆ν = √ (D − ν)2 + Cν . (A.6) There are altogether four solutions to the two quadratic equations (A.5), which may be written as aν = 1 2 + ε\|pν \|, bν = − ( D − ν − 3 2 ) + ε′∆ν+1, ε, ε′ = ±1. (A.7) Consequently, we get α = 1 2(1 + ε\|pν \|+ ε′∆ν+1 + ∆ν), β = 1 2(1 + ε\|pν \|+ ε′∆ν+1 −∆ν), γ = 1 + ε\|pν \|. (A.8) The general solution of the differential equation (A.4) may be written down as Fν(uν) = A 2F1(α, β; γ;uν) +Bu1−γν 2F1(α− γ + 1, β − γ + 1; 2− γ;uν), where A and B are two constants to be determined so that the angular function Θν(θν) be physically acceptable, i.e., vanish for θν → 0 and θν → π 2 . On considering the four possibilities for the pair (ε, ε′) in (A.7) and (A.8) successively, we arrive at a single solution corresponding either to ε = +1, ε′ = +1, B = 0, β = −nν (nν ∈ N) or to ε = −1, ε′ = +1, A = 0, β − γ + 1 = −nν (nν ∈ N). It can be expressed in terms of a Jacobi polynomial [84] as in equation (2.7), where aν = \|pν \|+ 1 2 , bν = −(D − ν − 3 2) + ∆ν+1, nν = 0, 1, 2, . . . , (A.9) while the separation constant Cν must satisfy the equation ∆ν = 2nν + \|pν \|+ ∆ν+1 + 1 (A.10) with ∆ν defined in (A.6). To obtain a solution to the whole set of D − 1 angular equations (A.1), as expressed in equation (2.6), it only remains to solve the recursion relation (A.10) for ∆ν with the starting value ∆D = \|pD\| corresponding to (A.3). The results for ∆ν and Cν , ν = 1, 2, . . . , D − 1, read ∆ν = 2nν + 2nν+1 + · · ·+ 2nD−1 + \|pν \|+ \|pν+1\|+ · · ·+ \|pD\|+D − ν and Cν = (2nν + 2nν+1 + · · ·+ 2nD−1 + \|pν \|+ \|pν+1\|+ · · ·+ \|pD\|) × (2nν + 2nν+1 + · · ·+ 2nD−1 + \|pν \|+ \|pν+1\|+ · · ·+ \|pD\|+ 2D − 2ν), (A.11) respectively. As a consequence, bν in (A.9) can be rewritten as in equation (2.8). This completes the proof of equations (2.6) to (2.8). 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Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

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