Intertwining Symmetry Algebras of Quantum Superintegrable Systems

We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n),so(2n)) or (su(p,q),so(2p,2q)). The eigenstates of the associated Hamiltonian hierarc...

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Date:	2009
Main Authors:	Calzada, J.A., Negro, J., del Olmo, M.A.
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Language:	English
Published:	Інститут математики НАН України 2009
Series:	Symmetry, Integrability and Geometry: Methods and Applications
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Cite this:	Intertwining Symmetry Algebras of Quantum Superintegrable Systems / J.A. Calzada, J. Negro, M.A. del Olmo // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 29 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1491672025-02-09T20:51:40Z Intertwining Symmetry Algebras of Quantum Superintegrable Systems Calzada, J.A. Negro, J. del Olmo, M.A. We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n),so(2n)) or (su(p,q),so(2p,2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness. This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with NonHermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). This work has been partially supported by DGES of the Ministerio de Educaci´on y Ciencia of Spain under Project FIS2005-03989 and Junta de Castilla y Le´on (Spain) (Project GR224). 2009 Article Intertwining Symmetry Algebras of Quantum Superintegrable Systems / J.A. Calzada, J. Negro, M.A. del Olmo // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 29 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B80; 81R12; 81R15 https://nasplib.isofts.kiev.ua/handle/123456789/149167 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n),so(2n)) or (su(p,q),so(2p,2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness.
format	Article
author	Calzada, J.A. Negro, J. del Olmo, M.A.
spellingShingle	Calzada, J.A. Negro, J. del Olmo, M.A. Intertwining Symmetry Algebras of Quantum Superintegrable Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Calzada, J.A. Negro, J. del Olmo, M.A.
author_sort	Calzada, J.A.
title	Intertwining Symmetry Algebras of Quantum Superintegrable Systems
title_short	Intertwining Symmetry Algebras of Quantum Superintegrable Systems
title_full	Intertwining Symmetry Algebras of Quantum Superintegrable Systems
title_fullStr	Intertwining Symmetry Algebras of Quantum Superintegrable Systems
title_full_unstemmed	Intertwining Symmetry Algebras of Quantum Superintegrable Systems
title_sort	intertwining symmetry algebras of quantum superintegrable systems
publisher	Інститут математики НАН України
publishDate	2009
url	https://nasplib.isofts.kiev.ua/handle/123456789/149167
citation_txt	Intertwining Symmetry Algebras of Quantum Superintegrable Systems / J.A. Calzada, J. Negro, M.A. del Olmo // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT calzadaja intertwiningsymmetryalgebrasofquantumsuperintegrablesystems AT negroj intertwiningsymmetryalgebrasofquantumsuperintegrablesystems AT delolmoma intertwiningsymmetryalgebrasofquantumsuperintegrablesystems
first_indexed	2025-11-30T16:09:55Z
last_indexed	2025-11-30T16:09:55Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 039, 23 pages Intertwining Symmetry Algebras of Quantum Superintegrable Systems? Juan A. CALZADA †, Javier NEGRO ‡ and Mariano A. DEL OLMO ‡ † Departamento de Matemática Aplicada, Universidad de Valladolid, E-47011, Valladolid, Spain E-mail: juacal@eis.uva.es ‡ Departamento de F́ısica Teórica, Universidad de Valladolid, E-47011, Valladolid, Spain E-mail: jnegro@fta.uva.es, olmo@fta.uva.es Received November 14, 2008, in final form March 18, 2009; Published online April 01, 2009 doi:10.3842/SIGMA.2009.039 Abstract. We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n), so(2n)) or (su(p, q), so(2p, 2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness. Key words: superintegrable systems; intertwining operators; dynamical algebras 2000 Mathematics Subject Classification: 17B80; 81R12; 81R15 1 Introduction It is well known that a Hamiltonian system (HS) in a configuration space of dimension n, is said to be integrable if there are n constants of motion, including the Hamiltonian H, which are independent and in involution. If the systems has 0 < k ≤ n− 1 additional constants of motion then, it is called superintegrable. The physical system is said to be maximally superintegrable if there exist 2n − 1 invariants well defined in phase-space. The superintegrable Hamiltonian systems (SHS) share nice properties. For instance, they admit separation of variables in more than one coordinate system for the Hamilton–Jacobi equation in the classical case and for the Schrödinger equation in the quantum case. Let us mention also that the finite classical trajectories are closed (periodic), while the discrete energy levels are degenerate in the quantum case. There is a limited number of this kind of physical systems as can be found in the works by Evans [1]. More recently, we quote the deformed algebra approach to superintegrability by Daskaloyannis and collaborators [2, 3] and the superintegrability in constant curvature configuration spaces [4, 5, 6]. Among a long list of contributions we can also mention two former references. In 1975 Lakshmanan and Eswaran [7] analyzed the isotropic oscillator on a 3-sphere and in 1979 motived by this work Higgs [8] studied versions of the Coulomb potential and of the harmonic oscillator living in the N -dimensional sphere and having SO(N + 1) and SU(N) symmetry in classical and in quantum mechanics, respectively. ?This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with Non- Hermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/PHHQP2008.html mailto:juacal@eis.uva.es mailto:jnegro@fta.uva.es mailto:olmo@fta.uva.es http://dx.doi.org/10.3842/SIGMA.2009.039 http://www.emis.de/journals/SIGMA/PHHQP2008.html 2 J.A. Calzada, J. Negro and M.A. del Olmo Some years ago a new family of SHS, was constructed from a group-theoretical method based on the symmetry reduction. These systems come, using the Marsden–Weinstein reduction [9], from free systems in Cp,q presenting an initial U(p, q)-symmetry [10] H = c 4 gµ̄ν p̄µpν MW reduction−→ Hr = c 4 gµνpsµpsν + V (s), where the bar stands for the complex conjugate and V (s) is a potential in terms of the real coordinates (sµ). These SHS are living in configuration spaces of constant curvature (SO(p, q)- homogeneous spaces). Although the quantum version of these systems is well known and can be exhaustively studied in all its aspects with standard procedures [10, 11, 12, 13, 14], we present here a new perspective based on intertwining operators (IO), a form of Darboux transformations [15], that will allow us to study them from an algebraic point of view. The associated IO’s close Lie algebras that take into account the symmetry properties of the systems and permit to describe these SHS in terms of representations of such “intertwining symmetry” algebras (or “dynamical algeb- ras” [16]). The intertwining operators are first order differential operators, A, connecting different Hamiltonians, H, H ′, in the same hierarchy, i.e., AH = H ′A. In the cases under study it is obtained a complete set of such IO’s, in the sense that any of the Hamiltonians of the hierar- chy can be expressed in terms of these operators. As we will see later, the IO’s are associated to systems of separable coordinates for the Hamiltonians. The study of the IO’s associated to integrable Hamiltonians has been made, for instance, in [17, 18, 19] and, following this line of re- search, we will supply here other non-trivial applications by means of the above mentioned family of SHS. From the perspective of the IO’s, we present a natural extension to higher dimensions of the intertwining (Darboux) transformations of the Schrödinger equation for one-dimensional quantum systems [20]. When a system of separable coordinates is used, any Hamiltonian of this family of SHS gives rise to a coupled set of n-differential equations, which can be factorized one by one. In principle, we will present two particular SHS that we denote u(3)-system [21] and u(2, 1)- system [22], but the generalization to higher u(p, q)-systems is evident. They are living in configuration spaces of constant curvature (SO(3) and SO(2, 1)-homogeneous spaces, respec- tively): a 2D sphere and a 2D hyperboloid of two-sheets. By extending well known methods in one-dimension to higher-dimensional systems we obtain a wide set of IO’s closing the dynamical Lie algebras u(3) or u(2, 1). These initial intertwining symmetry algebras can be enlarged by considering discrete symmetry operators obtaining, respectively, the so(6) and so(4, 2) Lie al- gebras of IO’s. This approach gives a simple explanation of the main features of these physical systems. For instance, it allows us to characterize the discrete spectrum and the corresponding eigenfunctions of the system by means of (finite/infinite) irreducible unitary representations (IUR) of the (compact/non-compact) intertwining symmetry algebras. We can compute the ground state and characterize the representation space of the wave-functions which share the same energy. The organization of the paper is as follows. In Section 2 we introduce the classical superintegrable Hamiltonian family under consideration. In Section 3 we focus on the quantum systems and show how to build the IO’s connecting hierarchies of these kind of Hamiltonians. It is seen that these operators close a su(2, 1) or a su(3) Lie algebra. The Hamiltonians are related to the second order Casimirs of such algebras, while the discrete spectrum of the Hamiltonians is related to their IUR’s. Next, a broader class of IO’s is defined leading to the so(4, 2) or so(6) Lie algebras, and it is shown how this new structure helps us to understand better the Hamiltonians in the new hierarchies. Finally, some remarks and conclusions in Section 4 will end the paper. Intertwining Symmetry Algebras of Quantum Superintegrable Systems 3 2 Superintegrable SU(p, q)-Hamiltonian systems Let us consider the free Hamiltonian H = 4 gµν̄pµp̄ν , µ, ν = 0, . . . , n = p+ q − 1, (1) (by pµ we denote the conjugate momenta) defined in the configuration space SU(p, q) SU(p− 1, q)× U(1) , which is an Hermitian hyperbolic space with metric gµν and coordinates yµ ∈ C such that gµ̄ν ȳ µyν = 1. The geometry and properties of this kind of spaces are described in [23] and [12]. Using a maximal Abelian subalgebra (MASA) of su(p, q) [24] the reduction procedure allows us to obtain a reduced Hamiltonian, which is not free, lying in the corresponding reduced space, a homogeneous SO(p, q)-space [10, 11] H = 1 2 gµνpsµpsν + V (s), where V (s) is a potential depending on the real coordinates sµ satisfying gµνsµsν = 1. The set of complex coordinates yµ after the reduction procedure becomes a set of ignorable variables xµ and the actual real coordinates sµ. A way to implement the symmetry reduction is as follows. Let Yµ, µ = 0, . . . , n, be a basis of the considered MASA of u(p, q) constituted only by pure imaginary matrices (this is a basic hypothesis in the reduction procedure). Then the relation between old (yµ) and new coordinates (xµ, sµ) is yµ = B(x)µνs ν , B(x) = exp (xµYµ) . The fact that the (xµ) are the parameters of the transformation associated to the MASA of u(p, q) used in the reduction, assures the ignorability of the x coordinates (in other words, the vector fields corresponding to the MASA are straightened out in these coordinates). The Jacobian matrix, J , corresponding to the coordinate transformation ((y, ȳ) → (x, s)) is given explicitly by J = ∂(y, ȳ) ∂(x, s) = ( A B Ā B̄ ) , where Aµν = ∂yµ ∂xν = (Yν) µ ρ y ρ. The expression of the Hamiltonian (1) in the new coordinates s is H = c ( 1 2 gµνpµpν + V (s) ) , V (s) = pTx (A†KA)−1px, where px are the constant momenta associated to the ignorable coordinates x and K is the matrix defined by the metric g. 4 J.A. Calzada, J. Negro and M.A. del Olmo 2.1 A classical superintegrable u(3)-Hamiltonian To obtain the classical superintegrable Hamiltonian associated to su(3), using the reduction procedure sketched before, we proceed as follows: let us consider the basis of su(3) determined by 3× 3 matrices X1, . . . , X8, whose explicit form, using the metric K = diag(1, 1, 1), is X1 = i 0 0 0 −i 0 0 0 0  , X2 = 0 0 0 0 i 0 0 0 −i  , X3 =  0 1 0 −1 0 0 0 0 0  , X4 = 0 i 0 i 0 0 0 0 0  , X5 =  0 0 1 0 0 0 −1 0 0  , X6 = 0 0 i 0 0 0 i 0 0  , X7 = 0 0 0 0 0 1 0 −1 0  , X8 = 0 0 0 0 0 i 0 i 0  . There is only one MASA for su(3): the Cartan subalgebra, generated by the matrices diag(i,−i, 0), diag(0, i,−i). So, we can generate only one su(3)-Hamiltonian system. In order to facilitate the computations we shall use the following basis for the corresponding MASA in u(3) Y0 = diag(i, 0, 0), Y1 = diag(0, i, 0), Y2 = diag(0, 0, i). (2) The actual real coordinates s are related to the complex coordinates y by yµ = sµe ixµ , µ = 0, 1, 2, and the Hamiltonian can be written as H = 1 2 ( p2 0 + p2 1 + p2 2 ) + V (s), V (s) = m2 0 s20 + m2 1 s21 + m2 2 s22 , (3) which lies in the 2-sphere (s0)2 + (s1)2 + (s2)2 = 1, with m0,m1,m2 ∈ R. The system is superintegrable since there exist three invariants of motion Rµν = (sµpν − sνpµ)2 + ( mµ sν sµ +mν sµ sν )2 , µ < ν, µ = 0, 1, ν = 1, 2. The constants of motion Rµν can be written in terms of the basis of su(3) (in the realization as function of sµ and pµ) Q1 ≡ R01 = X2 3 +X2 4 , Q2 ≡ R02 = X2 5 +X2 6 , Q3 ≡ R12 = X2 7 +X2 8 , and the sum of these invariants is the Hamiltonian (3) up to an additive constant H = Q1 +Q2 +Q3 + cnt. The quadratic Casimir of su(3) can be also written in terms of the constants of motion and the second order operators in the enveloping algebra of the compact Cartan subalgebra of su(3) Csu(3) = 3Q1 + 3Q2 + 3Q3 + 4X2 1 + 2[X1, X2]+ + 4X2 2 . The Hamiltonian is in involution with all the three constants of motion, i.e. [H,Qi] = 0, i = 1, 2, 3. However, the Qi’s do not commute among them [Q1, Q2] = [Q3, Q1] = [Q2, Q3] = −[X3, [X5, X7]+]+ − [X3, [X6, X8]+]+ + [X4, [X5, X8]+]+ − [X4, [X6, X7]+]+. So, the system (3) is superintegrable. Intertwining Symmetry Algebras of Quantum Superintegrable Systems 5 2.1.1 The Hamilton–Jacobi equation for the u(3)-system The solutions of the motion problem for this system can be obtained solving the corresponding Hamilton–Jacobi (HJ) equation in an appropriate coordinate system, such that the HJ equation separates into a system of ordinary differential equations. The 2-sphere can be parametrized on spherical coordinates (φ1, φ2) around the s2 axis by s0 = cosφ2 cosφ1, s1 = cosφ2 sinφ1, s2 = sinφ2, where φ1 ∈ [0, 2π) and φ2 ∈ [π/2, 3π/2]. Then, the Hamiltonian (3) is rewritten as H = 1 2 ( p2 φ2 + p2 φ1 cos2 φ2 ) + 1 cos2 φ2 ( m2 0 cos2 φ1 + m2 1 sin2 φ1 ) + m2 2 sin2 φ2 . The potential is periodic and has singularities along the coordinate lines φ1 = 0, π/2, π, 3π/2 and φ2 = π/2, 3π/2, and there is a unique minimum inside each domain of regularity. The invariants Qi can be rewritten in spherical coordinates taking the explicit form Q1 = 1 2 p2 φ1 + m2 0 cos 2 φ1 + m12 sin 2 φ1, Q2 = tan2 φ2 ( 1 2 p2 φ1 sin2 φ1 + m2 0 cos 2 φ1 ) + cos2 φ1 ( 1 2 p2 φ2 + m2 2 tan 2 φ2 ) + 1 2 pφ1pφ2 sin 2φ1 tanφ2, Q3 = tan2 φ2 ( 1 2 p2 φ1 cos2 φ1 + m2 1 sin 2 φ1 ) + sin2 φ1 ( 1 2 p2 φ2 + m2 2 tan 2 φ2 ) − 1 2 pφ1pφ2 sin 2φ1 tanφ2. Now, the HJ equation takes the form 1 2 ( ∂S ∂φ2 )2 + m2 2 sin2 φ2 + 1 cos2 φ2 ( 1 2 ( ∂S ∂φ1 )2 + m2 0 cos2 φ1 + m2 1 sin2 φ1 ) = E. It separates into two ordinary differential equations taking into account that the solution of the HJ equation can be written as S(φ1, φ2) = S1(φ1) + S2(φ2)− Et. Thus, 1 2 ( ∂S1 ∂φ1 )2 + m2 0 cos2 φ1 + m2 1 sin2 φ1 = α1, 1 2 ( ∂S2 ∂φ2 )2 + m2 2 sin2 φ2 + α1 cos2 φ2 = α2, where α2 = E and α1 are the separation constants (which are positive). Each one of these two equations is formaly similar to those of the corresponding one-dimensional problem [13]. The solutions of both HJ equations are easily computed and can be found as particular cases in [12]. Notice that all the orbits in a neighborhood of a critical point (center) are closed and, hence, the corresponding trajectories are periodic. The explicit solutions, when we restrict us to the domain 0 < φ1, φ2 < π/2, are cos2 φ2 = 1 2E [ b2 + √ b22 − 4α1E cos 2 √ 2Et ] , 6 J.A. Calzada, J. Negro and M.A. del Olmo cos2 φ1 = 1 2α1 [ b1 + 1 cos2 φ2 [ b21 − 4α1m 2 0 b22 − 4α1E ]1/2 ( (b2 cos2 φ2 − 2α1) sin 2 √ 2α1β1 + 2 √ α1 [ (b2 − E cos2 φ2) cos2 φ2 − α1 ]1/2 cos 2 √ 2α1β1 )] , where b1 = α1 + m2 0 − m2 1 and b2 = E + α1 − m2 2. Inside the domain the minimum for the potential is at the point (φ1 = arctan √ m1/m0, φ2 = arctan √ m2/(m0 +m1)), and its value is Vmin = (m0 +m1 +m2)2. Hence, the energy E is bounded from below E ≥ (m0 +m1 +m2)2. 2.2 A classical superintegrable u(2, 1)-Hamiltonian In a similar way to the preceding case u(3) of Section 2.1, it is enough to find an appropriate basis of u(2, 1), for instance X1 = i 0 0 0 −i 0 0 0 0  , X2 = 0 0 0 0 i 0 0 0 −i  , X3 =  0 1 0 −1 0 0 0 0 0  , X4 = 0 i 0 i 0 0 0 0 0  , X5 = 0 0 1 0 0 0 1 0 0  , X6 =  0 0 i 0 0 0 −i 0 0  , X7 = 0 0 0 0 0 1 0 1 0  , X8 = 0 0 0 0 0 i 0 −i 0  , and to follow the same procedure. However, the Lie algebra su(2, 1) has four MASAS [24]: the compact Cartan subalgebra like u(3), the noncompact Cartan subalgebra, the orthogonally decomposable subalgebra and the nilpotent subalgebra. For our purposes in this work we will only consider the symmetry reduction by the compact Cartan subalgebra, although we could generate other three SHS with the remaining MASAs. It is also possible to use the same matrices of u(3) (2) to build up a basis of the compact Cartan subalgebra. We find the following reduced Hamiltonian H = 1 2 c ( −p2 0 − p2 1 + p2 2 ) + m2 0 s20 + m2 1 s21 − m2 2 s22 , (4) lying in the 2-dimensional two-sheet hyperboloid −s20 − s21 + s22 = 1 and with c a constant. The potential constants, mi, can be chosen non-negative real numbers. Parametrizing the two-sheet hyperboloid by using an ‘analogue’ of the spherical coordinates s0 = sinh ξ cos θ, s1 = sinh ξ sin θ, s2 = cosh ξ, (5) with 0 ≤ θ < π/2 and 0 ≤ ξ <∞, the Hamiltonian can be rewritten if c = −1 as H = 1 2 ( p2 ξ + p2 θ sinh2 ξ ) + 1 sinh2 ξ ( m2 0 cos2 θ + m2 1 sin2 θ ) − m2 2 cosh2 ξ . The potential is regular inside the domain of the variables and there is a saddle point for the values θ = arctan √ m1/m0 and ξ = arg tanh √ m2(m0 +m1)) if m0 +m1 > m2. The quadratic constants of motion in terms of the enveloping algebra of su(2, 1) are Q1 = X2 3 +X2 4 , Q2 = X2 5 +X2 6 , Q3 = X2 7 +X2 8 , and the sum of these invariants gives also the Hamiltonian up to an additive constant H = −Q1 +Q2 +Q3 + cnt. Intertwining Symmetry Algebras of Quantum Superintegrable Systems 7 The quadratic Casimir of su(2, 1) is Csu(2,1) = 3Q1 − 3Q2 − 3Q3 + 4X2 1 + 2[X1, X2]+ + 4X2 2 . The Hamiltonian is in involution with all the three constants of motion that do not commute among themselves [Q1, Q2] = [Q3, Q1] = [Q3, Q2] = −[X3, [X5, X7]+]+ − [X3, [X6, X8]+]+ + [X4, [X5, X8]+]+ − [X4, [X6, X7]+]+. The explicit form of the invariants of motion in terms of the coordinates ξ and θ is Q1 = 1 2 p2 θ + m2 0 cos2 θ + m2 1 sin2 θ , Q2 = coth2 ξ ( 1 2 p2 θ sin2 θ + m2 0 cos2 θ ) + cos2 θ ( 1 2 p2 ξ + m2 2 coth2 ξ ) + 1 2 pθpξ sin 2θ coth ξ, Q3 = coth2 ξ ( 1 2 p2 θ cos2 θ + m2 1 sin2 θ ) + sin2 θ ( 1 2 p2 ξ + m2 2 coth2 ξ ) − 1 2 pθpξ sin 2θ coth ξ. 3 Superintegrable quantum systems In the previous Section 2 we have described some classical superintegrable systems. Now we will study their quantum versions. In order to construct the quantum version of both systems, let us proceed in the following way. By inspection of the classical Hamiltonians (3) and (4) and their constants of motions we can relate the terms like sµpν±sνpµ with generators of “rotations” when pµ → ∂µ in the plane XµXν . Moreover, since the Hamiltonian is the sum of the three constants of motion up to constants we can write a quantum Hamiltonian as a linear combination of J2 0 , J2 1 and J2 2 , being J0, J1 and J2 the infinitesimal generators of “rotations” in the plane X1X2, X0X2 and X0X1 around the axis X0, X1 and X2, respectively. According to the signature of the metric the rotations will be compact or noncompact, so, the generators will span so(3) or so(2, 1) and for our purposes we will take a differential realization of them. In other words, the Casimir operator of so(3) (Cso(3) = J2 0 +J2 1 +J2 2 ) or of so(2, 1) (Cso(2,1) = J2 0 +J2 1 −J2 2 ) gives the “kinetic” part of the corresponding Hamiltonian. In the following we will present a detailed study of the case related with su(2, 1). The su(3)-case is simply sketched and the interested reader can find more details in [21]. 3.1 Superintegrable quantum u(2, 1)-system Let us consider the Hamiltonian H` = −J2 0 − J2 1 + J2 2 + l20 − 1 4 s20 + l21 − 1 4 s21 − l22 − 1 4 s22 , (6) which configuration space is the 2-dimensional two-sheet hyperboloid −s20 − s21 + s22 = 1, with ` = (l0, l1, l2) ∈ R3 (and 2m = 1). The differential operators J0 = s1∂2 + s2∂1, J1 = s2∂0 + s0∂2, J2 = s0∂1 − s1∂0, constitute a realization of so(2, 1) with Lie commutators [J0, J1] = −J2, [J2, J0] = J1, [J1, J2] = J0. 8 J.A. Calzada, J. Negro and M.A. del Olmo Using coordinates (5) the explicit expressions of the infinitesimal generators are J0 = sin θ∂ξ + cos θ coth ξ∂θ, J1 = cos θ∂ξ − sin θ coth ξ∂θ, J2 = ∂θ. They are anti-Hermitian operators inside the space of square-integrable functions with invariant measure dµ(θ, ξ) = sinh ξ dθdξ. In these coordinates the Hamiltonian H` (6) has the expression H` = −∂2 ξ − coth ξ ∂ξ − l22 − 1 4 cosh2 ξ + 1 sinh2 ξ [ −∂2 θ + l21 − 1 4 sin2 θ + l20 − 1 4 cos2 θ ] . It can be separated in the variables ξ and θ, by choosing its eigenfunctions Φ` in the form Φ`(θ, ξ) = f(θ) g(ξ), obtaining a pair of separated equations Hθ l0,l1f(θ) ≡ [ −∂2 θ + l21 − 1 4 sin2 θ + l20 − 1 4 cos2 θ ] f(θ) = α f(θ), (7)[ −∂2 ξ − coth ξ∂ξ − l22 − 1 4 cosh2 ξ + α sinh2 ξ ] g(ξ) = Eg(ξ), where α > 0 is a separation constant. 3.1.1 A complete set of intertwining operators for H` The one-dimensional Hamiltonian Hθ l0,l1 (7) can be factorized as a product of first order opera- tors A± and a constant λ(l0,l1) Hθ (l0,l1) = A+ (l0,l1)A − (l0,l1) + λ(l0,l1), A±l0,l1 = ±∂θ − (l0 + 1/2)tan θ + (l1 + 1/2)cot θ, (8) λ(l0,l1) = (l0 + l1 + 1)2. The fundamental relation between contiguous couples of operators A±, Hθ (l0,l1) = A+ (l0,l1)A − (l0,l1) + λ(l0,l1) = A−(l0+1,l1+1)A + (l0+1,l1+1) + λ(l0+1,l1+1), allows us to construct a hierarchy of Hamiltonians . . . , Hθ l0−1,l1−1, H θ l0,l1 , H θ l0+1,l1+1, . . . , H θ l0+n,l1+n, . . . , which satisfy the recurrence relations A−(l0,l1)H θ (l0,l1) = Hθ (l0+1,l1+1)A − (l0,l1), A+ (l0,l1)H θ (l0+1,l1+1) = Hθ (l0,l1)A + (l0,l1). From the above relations we see that the operators A±(l0,l1) act as shape invariant intertwin- ing operators and also that A−(l0,l1) transforms eigenfunctions of Hθ (l0,l1) into eigenfunctions of Hθ (l0+1,l1+1), and viceversa for A+ (l0,l1), in such a way that the original and the transformed eigenfunctions have the same eigenvalue. Hence, once the initial values for (l0, l1) have been fixed we can build up an infinite set of Hamiltonians {Hθ (l0+n,l1+n)}n∈Z connected by the set of operators {A±(l0+n,l1+n)}n∈Z (Hamilto- nian ‘hierarchy’). Intertwining Symmetry Algebras of Quantum Superintegrable Systems 9 3.1.2 The u(2) ‘dynamical’ algebra We can define free-index operators Ĥθ, Â±, Â starting from the set of index-depending operators {Hθ (l0+n,l1+n), A ± (l0+n,l1+n)}n∈Z. The free-index operators act on the eigenfunctions f(l0+n,l1+n) of Hθ (l0+n,l1+n) as follows: Ĥθf(l0,l1) := Hθ (l0,l1)f(l0,l1), Â−f(l0,l1) := 1 2 A−(l0,l1)f(l0,l1), Â+f(l0+1,l1+1) := 1 2 A+ (l0,l1)f(l0+1,l1+1), Â f(l0,l1) := −1 2 (l0+l1)f(l0,l1). With this convention the free-index operators close a su(2)-algebra with commutators [Â, Â±] = ±A±, [Â+, Â−] = 2Â, (9) and including the operator Df(l0,l1) := (l0− l1)f(l0,l1), that commutes with the other three ones, we obtain a u(2)-algebra. The fundamental states of some distinguished Hamiltonians are in relation with the IUR’s of su(2). Thus, an eigenstate f0 (l0+n,l1+n) of Hθ (l0+n,l1+n) will be a fundamental (highest or lowest weight) vector if A−f0 (l0+n,l1+n) = A−(l0+n,l1+n)f 0 (l0+n,l1+n) = 0. The solution of this equation, f0 (l0+n,l1+n)(θ1) = N cosl0+1/2+n(θ1) sinl1+1/2+n(θ1) (10) with N a normalization constant and eigenvalue E0 (l0+n,l1+n) = λ(l0+n,l1+n) = (l0 + l1 + 1 + 2n)2, is also eigenfunction of A Af0 (l0+n,l1+n) = A(l0+n,l1+n)f 0 (l0+n,l1+n) = −1 2 (l0 + l1 + 2n)f0 (l0+n,l1+n). (11) The functions (10) are regular and square-integrable when l0, l1 ≥ −1/2. From (11) we can make the identification f0 (l0+n,l1+n) ' \|jn,−jn〉, with jn = 1 2(l0 + l1 + 2n), n = 0, 1, 2, . . . The representation, Djn , fixed by f0 (l0+n,l1+n) will be a IUR of su(2) of dimension 2jn+1 = l0+l1+2n+1 if l0+l1 ∈ Z+, n ∈ Z+. The Hamiltonian Hθ can be written in terms of the Casimir of su(2), C = A+A− +A(A− 1), as follows Hθ = 4(C + 1/4). The other eigenstates in the representation Djn are obtained applying recursively A+. Thus, fn(l0,l1) = (A+)nf0 (l0+n,l1+n) = A+ (l0,l1)A + (l0+1,l1+1) · · ·A + (l0+n−1,l1+n−1)f 0 (l0+n,l1+n), 10 J.A. Calzada, J. Negro and M.A. del Olmo and fn(l0,l1) ' \|jn,−jn + n〉. The explicit form of fn(l0,l1) is fn(l0,l1) = sinl1+1/2(φ1) cosl0+1/2(φ1)P (l1,l0) m [cos(2φ1)], being Pm the Jacobi polynomials, with eigenvalue En(l0,l1) = (l0 + l1 + 1 + 2n)2, n ∈ Z+. Therefore, the eigenstates of the hierarchy {H(l0+n,l1+n)}n∈Z when l0+l1 ∈ Z+ can be ‘organized’ in IUR’s of su(2) (or of u(2)). Notice that different fundamental states with values of l0 and l1, such that j0 = (l0 + l1)/2 is fixed, would lead to the same j-IUR of su(2), but different u(2)- IUR’s may correspond to states with the same energy (because Df(l0,l1) = (l0−l1)f(l0,l1)). Hence, these results push us to find a larger algebra of operators such that all the eigenstates with the same energy belong to only one of its IUR’s. Since the IO’s A±l0,l1 depend only on the θ-variable, they can act also as IO’s of the complete Hamiltonians H` (6) and its global eigenfunctions Φ`, leaving the parameter l2 unchanged A−`′H`′ = H`A − `′ , A+ `′H` = H`′A + `′ , where ` = (l0, l1, l2) and `′ = (l0 − 1, l1 − 1, l2). In this sense, many of the above relations can be straightforwardly extended under this global point of view. 3.1.3 Second set of pseudo-spherical coordinates A second coordinate set, obtained from the noncompact rotations around the axes s2 and s0 respectively, and that allows us to parametrize the hyperboloid and separate the Hamiltonian is the following one s0 = coshψ sinhχ, s1 = sinhψ, s2 = coshψ coshχ, with −∞ < ψ < +∞ and 0 ≤ χ < +∞. In these coordinates the so(2, 1)-generators take the expressions J0 = − tanhψ sinhχ∂χ + coshχ∂ψ, J1 = ∂χ, J2 = sinhχ∂ψ − tanhψ coshχ∂χ. The explicit expression of the Hamiltonian is now H` = −∂2 ψ − tanhψ∂ψ + l21 − 1 4 sinh2 ψ + 1 cosh2 ψ [ −∂2 χ + l20 − 1 4 sinh2 χ − l22 − 1 4 cosh2 χ ] . It can be separated in the variables ψ and χ considering the eigenfunctions H` of the form Φ(χ, ψ) = f(χ)g(ψ). Hence, we obtain the following two equations Hχ l0,l2 f(χ) ≡ [ −∂2 χ + l20 − 1 4 sinh2 χ − l22 − 1 4 cosh2 χ ] f(χ) = αf(χ), (12)[ −∂2 ψ − tanhψ∂ψ + l21 − 1 4 sinh2 ψ + α cosh2 ψ ] g(ψ) = Eg(ψ), with α a separation constant. Intertwining Symmetry Algebras of Quantum Superintegrable Systems 11 The Hamiltonian Hχ l0,l2 (12) can be factorized as a product of first order operators B± Hχ l0,l2 = B+ l0,l2 B− l0,l2 + λl0,l2 = B− l0−1,l2−1B + l0−1,l2−1 + λl0−1,l2−1, being B± l0,l2 = ±∂χ + (l2 + 1/2) tanhχ+ (l0 + 1/2) cothχ, λl0,l2 = −(1 + l0 + l2)2. In this case the intertwining relations take the form B− l0−1,l2−1H χ l0−1,l2−1 = Hχ l0,l2 B− l0−1,l2−1, B+ l0−1,l2−1H χ l0,l2 = Hχ l0−1,l2−1B + l0−1,l2−1. Hence, the operators B± connect eigenfunctions of Hχ l0,l2 in the following way B− l0−1,l2−1 : fl0−1,l2−1 → fl0,l2 , B+ l0−1,l2−1 : fl0,l2 → fl0−1,l2−1. The operators B± l0,l2 can be also expressed in terms of ξ and θ B± l0,l2 = ±(cos θ∂ξ − sin θ coth ξ∂θ) + (l2 + 1/2) tanh ξ cos θ + (l0 + 1/2) coth ξ sec θ. We also define new free-index operators B̂−fl0,l2 := 1 2 B− l0,l2 fl0,l2 , B̂+ fl0,l2 := 1 2 B+ l0,l2 fl0,l2 , B̂fl0,l2 := −1 2 (l0 + l2)fl0,l2 , that close a su(1, 1) Lie algebra [B̂+, B̂−] = −2 B̂, [B̂, B̂±] = ±B̂±. (13) Since su(1, 1) is non-compact, its IUR’s are infinite-dimensional. In this case we are interested in the discrete series having a fundamental state annihilated by the lowering operator B−f0 l0,l2 = 0. The explicit expression of these states is f0 l0,l2(χ) = N(coshχ)l2+1/2(sinhχ)l0+1/2, where N is a normalization constant. In order to have a regular and square-integrable function we impose that l0 ≥ −1/2, −k1 ≡ l0 + l2 < −1. Since B̂f0 l0,l2 = −1 2(l0 + l2)f0 l0,l2 , the lowest weight of this infinite-dimensional IUR of su(1, 1) is characterized by j′1 = k1/2 > 1/2. The IO’s B̂± can also be considered as intertwining operators of the Hamiltonians H` linking their eigenfunctions Φ`, similarly to the IO’s Â±, described before, but now with l1 remaining unchanged. 12 J.A. Calzada, J. Negro and M.A. del Olmo 3.1.4 Third set of pseudo-spherical coordinates A third set of coordinates is obtained from the noncompact rotations around the axes s1 and s0, respectively. It gives rise to the following parametrization of the hyperboloid s0 = sinhφ, s1 = coshφ sinhβ, s2 = coshφ coshβ, with 0 ≤ φ < +∞ and −∞ < β < +∞. The infinitesimal generators have the expressions J0 = ∂β, J1 = coshβ∂φ − tanhφ sinhβ ∂β, J2 = −sinhβ∂φ + tanhφ coshβ∂β . Hence, the Hamiltonian now takes the form H` = −∂2 φ − tanhφ∂φ + l20 − 1 4 sinh2 φ + 1 cosh2 φ [ −∂2 β + l21 − 1 4 sinh2 β − l22 − 1 4 cosh2 β ] , and it separates in the variables φ, β in terms of its eigenfunctions Φ(β, φ) = f(β)g(φ) Hβ l1,l2 f(β) ≡ [ −∂2 β + l21 − 1 4 sinh2 β − l22 − 1 4 cosh2 β ] f(β) = αf(β),[ −∂2 φ − tanhφ∂φ + l20 − 1 4 sinh2 φ + α cosh2 φ ] g(φ) = E g(φ), with the separation constant α. The second order operator Hβ l1,l2 can be factorized as a product of first order operators C± Hβ l1,l2 = C+ l1,l2 C− l1,l2 + λl1,l2 = C− l1+1,l2−1C + l1+1,l2−1 + λl1+1,l2−1, being C± l1,l2 = ±∂β + (l2 + 1/2) tanhβ + (−l1 + 1/2) cothβ, λl1,l2 = −(1− l1 + l2)2. The operators C± l1,l2 give rise to the intertwining relations C+ l1+1,l2−1H β l1,l2 = Hβ l1+1,l2−1C + l1+1,l2−1, C− l1+1,l2−1H β l1+1,l2−1 = Hβ l1,l2 C− l1+1,l2−1, which imply the connection among eigenfunctions C− l1+1,l2−1 : fl1+1,l2−1 → fl1,l2 , C+ l1+1,l2−1 : fl1,l2 → fl1+1,l2−1. The IO’s C± l1,l2 can also be expressed in terms of the first set of coordinates (ξ, θ) C± l1,l2 = ±(sin θ∂ξ + cos θ coth ξ∂θ) + (l2 + 1/2) tanh ξ sin θ + (−l1 + 1/2) coth ξ csc θ. New free-index operators are defined as Ĉ−fl1,l2 := 1 2 C− l1,l2 fl1,l2 , Ĉ+fl1,l2 := 1 2 C+ l1,l2 fl1,l2 , Ĉfl1,l2 := −1 2 (l2 − l1)fl1,l2 , satisfying the commutation relations of the su(1, 1) algebra [Ĉ−, Ĉ+] = 2 Ĉ, [Ĉ, Ĉ±] = ±Ĉ±. (14) Intertwining Symmetry Algebras of Quantum Superintegrable Systems 13 The fundamental state for the su(1, 1) representation, given by Ĉ−f0 l1,l2 = 0, has the expression f0 l1,l2(β) = N(coshβ)l2+1/2(sinhβ)−l1+1/2, with N a normalization constant. In order to get an IUR from this eigenfunction, we impose it to be regular and normalizable, therefore l1 ≤ 1/2, −k2 ≡ l2 − l1 < −1. The lowest weight of the IUR is given by j′2 = k2/2 > 1/2, because in this case we have that Ĉf0 l1,l2 = −1 2(l2 − l1)f0 l1,l2 . As in the other cases the IO’s C± can be considered as connecting global Hamiltonians H` and their eigenfunctions, having in mind that now the parameter l0 is unaltered. 3.1.5 Algebraic structure of the intertwining operators If we consider together all the IO’s {Â±, Â, B̂±, B̂, Ĉ±, Ĉ} we find that they close a su(2, 1) Lie algebra, whose Lie commutators are displayed in (9), (13) and (14) together with the crossed commutators [Â+, B̂+] = 0, [Â+, B̂−] = −Ĉ−, [Â+, B̂] = −1 2 Â+, [Â+, Ĉ+] = B̂+, [Â+, Ĉ−] = 0, [Â+, Ĉ] = 1 2 Â+, [Â−, B̂+] = Ĉ+, [Â−, B̂−] = 0, [Â−, B̂] = 1 2 Â−, [Â−, Ĉ+] = 0, [Â−, Ĉ−] = −B̂−, [Â−, Ĉ] = −1 2 Â−, [Â, B̂+] = 1 2 B̂+, [Â, B̂−] = −1 2 B̂−, [Â, B̂] = 0, [Â, Ĉ+] = −1 2 Ĉ+, [Â, Ĉ−] = 1 2 Ĉ−, [Â, Ĉ] = 0, [B̂+, Ĉ+] = 0, [B̂+, Ĉ−] = −Â+, [B̂+, Ĉ] = −1 2 B̂+, [B̂−, Ĉ+] = Â−, [B̂−, Ĉ−] = 0, [B̂−, Ĉ] = 1 2 B̂−, [B̂, Ĉ+] = 1 2 Ĉ+, [B̂, Ĉ−] = −1 2 Ĉ−, [B̂, Ĉ] = 0. The second order Casimir operator of su(2, 1) is C = Â+Â− − B̂+B̂− − Ĉ+Ĉ− + 2 3 ( Â2 + B̂2 + Ĉ2 ) − (Â+ B̂ + Ĉ). Note that in our differential realization Â− B̂ + Ĉ = 0, and that there is another generator, C′ = l1 + l2 − l0, 14 J.A. Calzada, J. Negro and M.A. del Olmo commuting with the rest of generators of su(2, 1). Hence 〈Â±, Â, B̂±, B̂, Ĉ±, Ĉ〉 ⊕ 〈C′〉 ≈ u(2, 1). The Hamiltonian (6) can be rewritten in terms of C and C′ as H` = −4C + 1 3 C′2 − 15 4 = −4 ( Â+Â− − B̂+B̂− − Ĉ+Ĉ− + 2 3 ( Â2 + B̂2 + Ĉ2 ) − (Â+ B̂ + Ĉ) ) + 1 3 C′2 − 15 4 . The quadratic operators Â+Â−, B̂+B̂− and Ĉ+Ĉ− commute with the Hamiltonian and they are constants of motion. However, they do not commute among themselves giving cubic expressions [Â+Â−, B̂+B̂−] = −[Â+Â−, Ĉ+Ĉ−] = −[B̂+B̂−, Ĉ+Ĉ−] = Â+Ĉ+B̂− − B̂+Ĉ−Â−. However, as we will see later, they generate a quadratic algebra. The eigenfunctions of the Hamiltonians H`, that have the same energy, support an IUR of su(2, 1) characterized by a value of C and other of C′. These representations can be obtained, as usual, starting from a fundamental state simultaneously annihilated by the lowering opera- tors Â−, Ĉ− and B̂− Â−` Φ0 ` = Ĉ− ` Φ0 ` = B̂− ` Φ0 ` = 0. Solving these equations we find Φ0 `(ξ, θ) = N(cos θ)l0+1/2(sin θ)1/2(cosh ξ)l2+1/2(sinh ξ)l0+1, (15) where ` = (l0, 0, l2) and N is a normalization constant. From previous inequalities the parame- ters l0, l2 of Φ0 ` must satisfy (l0 + l2) < −3/2, l0 ≥ −1/2. In order to guarantee the normalization of Φ0 ` using the invariant measure the values of the parameters l0 and l2 have to verify (l0 + l2) < −5/2. Note that the state Φ0 ` supports also IUR’s of the subalgebras su(2) (generated by Â± with the weight j = l0/2) and su(1, 1) (generated by Ĉ± with j′2 = −l2/2). The energies of the fundamental states Φ0 `(ξ, θ) are obtained from H` taking into account the expressions for the Casimir operators C and C′ H`Φ0 ` = −(l0 + l2 + 3/2)(l0 + l2 + 5/2)Φ0 ` ≡ E0 `Φ 0 ` . From Φ0 ` we can get the other eigenfunctions in the su(2, 1) representation using the raising operators Â+, B̂+, Ĉ+, all of them sharing the same energy eigenvalue E0 ` . Notice that the expression for E0 ` depends on l0 + l2, hence the states in the family of IUR’s derived from fundamental states Φ0 `(ξ, θ), sharing the same value of l0 + l2, also shall have the same energy eigenvalue. The energy E0 ` corresponding to bound states is negative and the set of such bound states for each Hamiltonian H` is finite. In Fig. 1 we display the states of some IUR’s of su(2, 1) by points (l0, l1, l2) ∈ R3 linked to the ground state Φ0 ` , characterized by (l0, 0, l2), by the raising operators Â+ and Ĉ+. The points associated to a IUR are in a 2D plane (fixed by the particular value l0 + l2 = −3 of C′) and, Intertwining Symmetry Algebras of Quantum Superintegrable Systems 15 Figure 1. States of IUR’s of su(2, 1) sharing the same energy and represented by points in the three dark planes associated to Φ0 ` with ` = (0, 0,−3), ` = (1, 0,−4) and ` = (2, 0,−5). obviously, the other IUR’s are described by points in parallel 2D planes. These parallel planes are placed inside a tetrahedral unbounded pyramid whose basis extends towards −∞ along the axis l2. On the other hand, there exist some points (in the parameter space of parameters (l0, l1, l2)) which are degenerated because they correspond to an eigenspace with dimension bigger than 1. For instance, let us consider the representation characterized by the fundamental state Φ0 ` where ` = (0, 0,−3): its points lie in a triangle and are nondegenerated. The IUR cor- responding to the ground state with `′ = (1, 0,−4) has eigenstates with the same energy, E = −(−3 + 3/2)(−3 + 5/2), as the previous one, since both share the same value of l0+l2 = −3. The eigenstates corresponding to `′′ = (0, 0,−5), inside this representation, may be obtained in two ways: Φ2 (0,0,−5) = Ĉ+Â+Φ0 (0,0,−3), Φ̃2 (0,0,−5) = Â+Ĉ+Φ0 (0,0,−3). We have two independent states spanning a 2-dimensional eigenspace of the HamiltonianH(0,0,−5) for that eigenvalue of the energy E = −(−3 + 3/2)(−3 + 5/2). The ground state for H(0,0,−5) is given by the wavefunction Φ0 (0,0,−5) and its energy is E0 (0,0,−5) = −(−5 + 3/2)(−5 + 5/2). In a similar way it is possible to obtain the degeneration of higher excited levels in the discrete spectrum of the Hamiltonians. Thus, the n-excited level, when it exists, has associated an n-dimensional eigenspace. 3.1.6 The complete symmetry algebra so(4, 2) By simple inspection one can see that the Hamiltonian H` (6) is invariant under reflections in the space of parameters (l0, l1, l2) I0 : (l0, l1, l2) → (−l0, l1, l2), I1 : (l0, l1, l2) → (l0,−l1, l2), I2 : (l0, l1, l2) → (l0, l1,−l2). (16) These operators generate by conjugation other sets of intertwining operators from the ones already defined. Thus, I0 : {Â±, Â} −→ {Ã± = I0Â ±I0, Ã = I0ÂI0}, 16 J.A. Calzada, J. Negro and M.A. del Olmo where Ã±l0,l1 = ±∂θ − (−l0 + 1/2) tan θ + (l1 + 1/2) cot θ, λ̃l0,l1 = (1− l0 + l1)2. They act on the eigenfunctions of the Hamiltonians (7) in the following way Ã−l0,l1 : fl0,l1 → fl0−1,l1+1, Ã+ l0,l1 : fl0−1,l1+1 → fl0,l1 . In these conditions, we can define global operators Ã± as we made before. Then, Ã± together with Ãfl0,l1 := −1 2(−l0 + l1)fl0,l1 close a second s̃u(2). In a similar way new sets of operators {B̃±, B̃} and {C̃±, C̃} closing s̃u(1, 1) algebras, can also be defined I0 : {A±, A;B±, B;C±, C} −→ {Ã±, Ã; B̃±, B̃;C±, C}, I1 : {A±, A;B±, B;C±, C} −→ {Ã∓,−Ã;B±, B; C̃±, C̃}, I2 : {A±, A;B±, B;C±, C} −→ {A±, A; B̃∓,−B̃;−C̃∓,−C̃}. The whole set of the operators {A±, Ã±, B±, B̃±, C±, C̃±} together with the set of diagonal operators {L0, L1, L2}, defined by LiΨ` = liΨ`, span a Lie algebra of rank three: o(4, 2). The Lie commutators of o(4, 2) can be easily derived from those of su(2, 1) and the action of the reflections. It is obvious, by construction, that all these generators link eigenstates of Hamiltonians H` with the same eigenvalue. The fundamental state Ψ0 ` for so(4, 2) will be annihilated by all the lowering operators A−` Ψ0 ` = Ã−` Ψ0 ` = C− ` Ψ0 ` = C̃− ` Ψ0 ` = B− ` Ψ0 ` = B̃− ` Ψ0 ` = 0. This state should be a particular case of the state given by expression (15), i.e. it should be also invariant under the l0-reflection, Φ0 (l0=0,l1=0,l2)(ξ, θ) = N(cos θ)1/2(sin θ)1/2(cosh ξ)l2+1/2 sinh ξ, where l2 < −5/2. This point (l0 = 0, l1 = 0, l2) in the parameter space, for the cases displayed in Fig. 1, corresponds to the top vertex of the pyramid, from which all the other points plotted can be obtained with the help of raising operators. Such points correspond to an IUR of so(4, 2) algebra that includes the series of IUR’s of su(2, 1). Fixed the IUR of so(4, 2) corresponding to a value of ` = (0, 0, l2) such that−7/2 ≤ l2 < −5/2, then the points on the surface of the associated pyramid in the parameter space correspond to non-degenerated ground levels of their respective Hamiltonians. This ‘top’ pyramid includes inside other ‘lower’ pyramids with vertexes at the points `n = (0, 0, l2 − 2n). Each point on the surface of an inner pyramid associated to `n represents an n-excited level n-fold degenerated of the IUR associated to ` (see Fig. 2). 3.2 Superintegrable quantum u(3)-system In this case we consider the quantum Hamiltonian H` = − ( J2 0 + J2 1 + J2 2 ) + l20 − 1/4 (s0)2 + l21 − 1/4 (s1)2 + l22 − 1/4 (s2)2 , (17) Intertwining Symmetry Algebras of Quantum Superintegrable Systems 17 Figure 2. Two pyramids associated to the same IUR of so(4, 2). The points of the faces of the exterior pyramid (with vertex (0, 0,−3)) represent non-degenerated levels. The exterior faces of the inner pyramid (vertex (0, 0,−5)) are first excited double-degenerated levels. where ` = (l0, l1, l2) ∈ R3, Ji = −εijksj∂k (i = 0, 1, 2) and its configuration space is the 2-sphere S2 ≡ (s0)2 + (s1)2 + (s2)2 = 1, (s0, s1, s2) ∈ R3. In spherical coordinates s0 = cosφ cos θ, s1 = cosφ sin θ, s2 = sinφ, φ ∈ [−π/2, π/2], θ ∈ [0, 2π], (18) that parametrize S2, the eigenvalue problem H`Ψ = EΨ takes the expression[ −∂2 φ + tanφ∂φ + l22 − 1/4 sin2(φ) + 1 cos2 φ [ −∂2 θ + l20 − 1/4 cos2 θ + l21 − 1/4 sin2 θ ]] Ψ = EΨ. Taking solutions separated in the variables θ and φ as Ψ(θ, φ) = f(θ)g(φ) we find Hθ l0,l1f(θ) ≡ [ −∂2 θ + l20 − 1/4 cos2 θ + l21 − 1/4 sin2 θ ] f(θ) = αf(θ), (19)[ −∂2 φ + tanφ∂φ + α cos2 φ + l22 − 1/4 sin2 φ ] g(φ) = Eg(φ), with α > 0 a separating constant. Note that equation (19) is equal to equation (7) corresponding to the su(2, 1) case. Following the procedure of the previous case of so(2, 1) we can factorize the Hamiltonian (19) in terms of operators A±n like those of expression (8), obtaining, finally, a su(2) algebra. We can find other two sets of spherical coordinates, that parametrize the sphere S2 and that separate the Hamiltonian (19), Thus, we get two new sets of intertwining operators B± n and C± n , like in the su(2, 1) case. In this way, we can construct an algebra u(3) and using reflection operators, acting in the space of the parameters of the Hamiltonian (17), this algebra is enlarged to so(6). These three sets of operators are related, as we saw in Section 3.1, with three sets of (spheri- cal) coordinates that we can take in the 2-sphere immersed in a 3-dimensional ambient space 18 J.A. Calzada, J. Negro and M.A. del Olmo with cartesian axes {s0, s1, s2}. Since the coordinates (s0, s1, s2) play a symmetric role in the Hamiltonian (17), we will take their cyclic rotations to get two other intertwining sets. Thus, we take the spherical coordinates choosing as ‘third axis’ s1 instead of s2 as in (18), s2 = cosψ cos ξ, s0 = cosψ sin ξ, s1 = sinψ, ψ ∈ [−π/2, π/2], ξ ∈ [0, 2π]. The corresponding intertwining operators B± (l0,l1,l2) are defined in a similar way to A±(l0,l1,l2). The explicit expressions for the new set in terms of the initial coordinates (θ, φ) (18) are B± (l0,l1,l2) = ±(sin θ tanφ∂θ + cos θ∂φ)− (l2+1/2) cos θ cotφ+ (l0+1/2) sec θ tanφ. The spherical coordinates around the s0 axis are s1 = cosβ cos η, s2 = cosβ sin η, s0 = sinβ. We obtain a new pair of operators, that written in terms of the original variables (θ, φ) are C± (l0,l1,l2) = ±(cos θ tanφ∂θ − sin θ∂φ) + (l1−1/2) cosec θ tanφ+ (l2+1/2) sin θ cotφ. They intertwine the Hamiltonians in the following way C− (l0,l1,l2)H(l0,l1,l2) = H(l0,l1−1,l2+1)C − (l0,l1,l2), C+ (l0,l1,l2)H(l0,l1−1,l2+1) = H(l0,l1,l2)C + (l0,l1,l2). The free-index or ‘global’ operators close a third su(2). All these transformations {A±, A,B±, B, C±, C} (where A − B + C = 0) span an algebra su(3), whose Lie commutators are [Â+, Â−] = 2A, [Â, Â±] = ±Â±, [B̂+, B̂−] = 2B, [B̂, B̂±] = ±B̂±, [Ĉ, Ĉ±] = ±Ĉ±, [Ĉ+, Ĉ−] = 2C, [Â+, B̂+] = 0, [Â+, B̂−] = Ĉ−, [Â+, B̂] = −1 2 Â+, [Â+, Ĉ+] = −B̂+, [Â+, Ĉ−] = 0, [Â+, Ĉ] = 1 2 Â+, [Â−, B̂+] = −Ĉ+, [Â−, B̂−] = 0, [Â−, B̂] = 1 2 Â−, [Â−, Ĉ+] = 0, [Â−, Ĉ−] = B̂−, [Â−, Ĉ] = −1 2 Â−, [Â, B̂+] = 1 2 B̂+, [Â, B̂−] = −1 2 B̂−, [Â, B̂] = 0, [Â, Ĉ+] = −1 2 Ĉ+, [Â, Ĉ−] = 1 2 Ĉ−, [Â, Ĉ] = 0, [B̂+, Ĉ+] = 0, [B̂+, Ĉ−] = −Â+, [B̂+, Ĉ] = −1 2 B̂+, [B̂−, Ĉ+] = Â−, [B̂−, Ĉ−] = 0, [B̂−, Ĉ] = 1 2 B̂−, [B̂, Ĉ+] = 1 2 Ĉ+, [B̂, Ĉ−] = −1 2 Ĉ−, [B̂, Ĉ] = 0. The second order Casimir operator of su(3) is given by C = A+A− +B+B− + C+C− + 2 3 A(A− 3/2) + 2 3 B(B − 3/2) + 2 3 C(C − 3/2). (20) We obtain a u(3) algebra by adding the central diagonal operator D := l0 − l1 − l2. (21) Intertwining Symmetry Algebras of Quantum Superintegrable Systems 19 The global operator convention can be adopted for the Hamiltonians in the u(3)-hierarchy by defining its action on the eigenfunctions Φ(l1,l2,l3) of H(l1,l2,l3) by HΦ(l1,l2,l3) := H(l1,l2,l3)Φ(l1,l2,l3). Then, H can be expressed in terms of both Casimir operators, (20) and (21), as H = 4C − 1 3 D2 + 15 4 . (22) Hence, the Hamiltonian can be written as a certain quadratic function of the operators A±, B± and C± generalizing the usual factorization for one-dimensional systems plus a constant since in the representation that we are using the operators A, B, C are diagonal depending on the parameters l0, l1, l2, H = 4(A+A− +B+B− + C+C−) + cnt. The quadratic operators A+A−, B+B−, C+C− commute with H but do not commute among themselves [A+A−, B+B−] = −[A+A−, C+C−] = [B+B−, C+C−] = −A+C+B− +B+C−A−. The intertwining operators can help also in supplying the elementary integrals of motion. We have two kinds of integrals: (i) second order constants, defined by the quadratic operators X1 = A+A, X2 = B+B, X3 = C+C; and (ii) third order constants defined by cubic operators: Y1 = A+C+B−, Y2 = B+C−A− = (Y1)+. Of course, since this system is superintegrable, there are only three functionally independent constants of motion, for instance X1, X2, X3. This set of symmetries {Xi, Yj} closes a quadratic algebra, as it is well known from many references [3, 18, 25, 26, 27, 28]. The commutators in this case are [X1, X2] = −[X1, X3] = [X2, X3] = −Y1 + Y2, [X1, Y1] = X1X2 −X1X3 − 2(A− 1)Y1, [X1, Y2] = −X2X1 +X3X1 + 2(A− 1)Y2, [X2, Y1] = X1X2 −X2X3 − (1 + 2B)Y1 + Y2 − 2CX2, [X2, Y2] = −X2X1 +X3X2 + (1 + 2B)Y2 − Y1 + 2CX2, [X3, Y1] = −X1X3 −X2X3 + 2CY1 − 2CX2 + Y2, [X3, Y2] = X3X1 +X3X2 − 2CY2 + 2CX2 − Y1, [Y1, Y2] = 2(−CX1X2 +BX1X3 +AX2X3 + (B + C)Y1 −AY2 + 2ACX2). Remark that the operators {A,B,C} are diagonal with fixed values for each Hamiltonian. One can show that the eigenstates of this Hamiltonian hierarchy are connected to the IUR’s of u(3). Fundamental states Φ annihilated by A− and C− (simple roots of su(3)), A−` Φ` = C− ` Φ` = 0, only exist when l1 = 0. Their explicit form is Φ`(θ, φ) = N cosl0+1/2 θ sin1/2 θ cosl0+1 φ sinl2+1/2 φ, whit N a normalizing constant. The diagonal operators act on them as AΦ` = −l0/2Φ`, l0 = m, l1 = 0, m = 0, 1, 2, . . . , CΦ` = −l2/2Φ`, l2 = n, n = 0, 1, 2, . . . . (23) This shows that Φ` is the lowest state of the IUR j1 = m/2 of the subalgebra su(2) generated by {A±, A}, and of the IUR j2 = n/2 of the subalgebra su(2) spanned by {C±, C}. Such 20 J.A. Calzada, J. Negro and M.A. del Olmo a su(3)-representation will be denoted (m,n), m,n ∈ Z≥0. The points (labelling the states) of this representation obtained from Φ` lie on the plane D = m− n inside the `-parameter space. The energy for the states of the IUR, determined by the lowest state (23) with parameters (l0, 0, l2), is given (22) by E = (l0 + l2 + 3/2)(l0 + l2 + 5/2) = (m+ n+ 3/2)(m+ n+ 5/2). Note that the IUR’s labelled by (m,n) with the same value m+ n are associated to states with the same energy (iso-energy representations). This degeneration will be broken using so(6). Making use of some relevant discrete symmetries, following the procedure of Section 3.1.6, the dynamical algebra u(3) can be enlarged to so(6). The Hamiltonian H(l0,l1,l2) (17) is invariant under reflections (16) in the parameter space (l0, l1, l2). These symmetries, Ii, can be directly implemented in the eigenfunction space, gi- ving by conjugation another set of intertwining operators {X̃ = IiXIi, i = 0, 1, 2} closing an isomorphic Lie algebra ũ(3). They are {A±, B±, C±} I0−→ {Ã∓, B̃∓, C±}, {A±, B±, C±} I1−→ {Ã±, B±, C̃±}, {A±, B±, C±} I2−→ {A±, B̃±, C̃∓}. The set {A±, Ã±, B±, B̃±, C±, C̃±, A, Ã, B, B̃, C, C̃} closes a Lie algebra of rank 3: so(6). How- ever, instead of the six non-independent generators A, Ã,B, B̃, C, C̃ it is enough to consider three independent diagonal operators L0, L1, L2 defined by LiΨ(l0,l1,l2) ≡ liΨ(l0,l1,l2). The Hamilto- nian can be expressed in terms of the so(6)-Casimir operator by means of the ‘symmetrization’ of the u(3)-Hamiltonian (22) H = {A+, A−}+ {B+, B−}+ {C+, C−}+ {Ã+, Ã−}+ {B̃+, B̃−}+ {C̃+, C̃−} + L0 2 + L1 2 + L2 2 + 41 12 . The intertwining generators of so(6) give rise to larger 3-dimensional Hamiltonian hierarchies {H(l0+m+p,l1+m−n−p,l2+n)}, m, n, p ∈ Z, (24) each one including a class of the previous ones coming from u(3). The eigenstates of these so(6)-hierarchies can be classified in terms of so(6) representations whose fundamental states Φ0 ` are determined by A−Φ0 ` = Ã−Φ0 ` = C−Φ0 ` , and whose explicit expressions are Φ0 `(θ, φ) = N cos1/2 θ sin1/2 θ cosφ sinl2+1/2 φ. They are characterized by the eigenvalues of the diagonal operators Li L0Φ0 ` = L1Φ0 ` = 0, L2Φ0 ` = nΦ0 ` , n ∈ Z+. We obtain two classes of symmetric IUR,s of so(6) that according to (24) we summarize as even IURs : l0 = 0, l1 = 0, l2 = 0, {H(m+p,m−n−p,n)}, qm, n, p ∈ Z, odd IURs : l0 = 0, l1 = 0, l2 = 1, {H(m+p,m−n−p,1+n)}, m, n, p ∈ Z. Intertwining Symmetry Algebras of Quantum Superintegrable Systems 21 Figure 3. The points represent the states of two IUR’s of so(6) with q = 1 (left) and q = 3 (right). The points corresponding to q = 3 include those of q = 1 (the inner octahedron) which are double degenerated. Figure 4. (Left) q = 1 IUR of so(6) where the triangular opposite faces correspond to two IUR’s of su(3). (Right) Points of a q = 2 IUR of so(6) that are associated to three IUR’s of su(3). These representations depicted in the parameter space correspond to octahedrons, that con- tain iso-energy representations of su(3) labeled by (m,n) such that m + n = q is fixed. In Fig. 3 we represent two IUR’s of so(6) characterized by q = 1 and q = 3, respectively. The 6 (q = 1)-eigenstates have energy E = 5 2 · 7 2 and the 50 (q = 3)-eigenstates share energy E = 7 2 · 9 2 . In Fig. 4 we show how the IUR’s of so(6) corresponding to q = 1 and q = 2 include IUR’s of su(3). Thus, in the case q = 1 the (m,n)-IUR’s involved are (1, 0) and (0, 1): the two parallel exterior faces of the octahedron faces that lie on the plane characterized by m−n. In the other case q = 2, the su(3)-IUR’s are (2, 0), (1, 1) and (0, 2). The first and the last ones correspond to the opposite parallel faces of the octahedron, the (1, 1)-IUR is represented in the parallel hexagonal section containing the origin. 4 Conclusions In this section we will enumerate a list of interesting features coming from the analysis of the IO’s associated to a SHS, for instance, by means of the example defined on a two-sheet hyperboloid. Obviously, from the Hamiltonian living in the sphere we can arrive to the same conclusions. 22 J.A. Calzada, J. Negro and M.A. del Olmo The IO’s of a SHS close an algebraic structure, in this case a non-compact su(2, 1) Lie al- gebra. By using the reflections operators of the system we can implement these IO’s obtaining a broader algebra: so(4, 2). These IO’s lead to hierarchies of Hamiltonians described by points on planes (su(2, 1)) or in the 3-dimensional space (so(4, 2)), corresponding to the rank of the re- spective algebra. This framework of IO’s can be very helpful in the characterization of a physical system by selecting separable coordinate systems and determining the eigenvalues and building eigenfunctions. We have shown the relation of eigenstates and eigenvalues with unitary representations of the su(2, 1) and so(4, 2) Lie algebras. In particular, we have studied the degeneration problem as well as the number of bound states. Here, we remark that such a detailed study for a ‘non- compact’ superintegrable system had not been realized till now, up to our knowledge. We have restricted to IUR’s, but a wider analysis can be done for hierarchies associated to representations with a not well defined unitary character. The IO’s can also be used to find the second order integrals of motion for a Hamiltonian H` and their algebraic relations, which is the usual approach to (super)-integrable systems. How- ever, we see that it is much easier to deal directly with the IO’s, which are more elementary and simpler, than with constants of motion. The second and third order constants of motion close a quadratic algebra. By means of the IO technique we have recovered the algebraic structure of the system that was used in the Marsden–Weinstein reducing procedure su(p, q) M−W−→ so(p, q) factoriz.−→ u(p, q) discrete−→ so(2p, 2q). This is a step to confirm the conjecture by Grabowski–Landi–Marmo–Vilasi [29] that “any completely integrable system should arise as reduction of a simpler one (associated for instance to a simple Lie algebra)”. Our program in the near future is the application of this method to wider situations. For example, to be useful when dealing with other SHS, but not necessarily maximally integrable, or not having a system of separable variable but still allowing algebraic methods. Acknowledgments This work has been partially supported by DGES of the Ministerio de Educación y Ciencia of Spain under Project FIS2005-03989 and Junta de Castilla y León (Spain) (Project GR224). References [1] Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666–5676. Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483–486. Evans N.W., Group theory of the Smorodinsky–Winternitz system, J. Math. Phys. 32 (1991), 3369–3375. 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Phys. 42 (1994), 393–427, hep-th/9307018. http://arxiv.org/abs/math-ph/0504016 http://arxiv.org/abs/hep-th/9407080 http://arxiv.org/abs/solv-int/9810010 http://arxiv.org/abs/quant-ph/0111034 http://arxiv.org/abs/quant-ph/0201099 http://arxiv.org/abs/math-ph/0601067 http://arxiv.org/abs/0803.2117 http://arxiv.org/abs/hep-th/9307018 1 Introduction 2 Superintegrable SU(p,q)-Hamiltonian systems 2.1 A classical superintegrable u(3)-Hamiltonian 2.1.1 The Hamilton-Jacobi equation for the u(3)-system 2.2 A classical superintegrable u(2,1)-Hamiltonian 3 Superintegrable quantum systems 3.1 Superintegrable quantum u(2,1)-system 3.1.1 A complete set of intertwining operators for H_l 3.1.2 The u(2) 'dynamical' algebra 3.1.3 Second set of pseudo-spherical coordinates 3.1.4 Third set of pseudo-spherical coordinates 3.1.5 Algebraic structure of the intertwining operators 3.1.6 The complete symmetry algebra so(4,2) 3.2 Superintegrable quantum u(3)-system 4 Conclusions References

Intertwining Symmetry Algebras of Quantum Superintegrable Systems

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