Table Of Content

Dynamical symmetries for superintegrable quantum systems J.A. Calzada 1, J. Negro 2, M.A. del Olmo 2 1 Departamento de Matema´tica Aplicada, 6 0 2 Departamento de F´ısica Teo´rica, 0 Universidad de Valladolid, E-47011, Valladolid, Spain 2 n E. mail: [email protected], [email protected], [email protected] a J 1 3 ] I S Abstract . n i Westudythedynamicalsymmetriesofaclassoftwo-dimensionalsuperintegrablesys- l n tems on a 2-sphere,obtained by a procedure based on the Marsden-Weinstein reduction, [ by considering its shape-invariant intertwining operators. These are obtained by gener- 1 alizing the techniques of factorization of one-dimensional systems. We firstly obtain a v pairofnoncommutingLiealgebrassu(2)thatoriginatethealgebraso(4). Byconsidering 9 three spherical coordinate systems we get the algebra u(3) that can be enlarged by ‘re- 6 flexions’to so(6). The bounded eigenstatesofthe Hamiltonianhierarchiesareassociated 0 to the irreducible unitary representations of these dynamical algebras. 1 0 6 0 1 Introduction / n i l It is well known that integrable Hamiltonian systems play a fundamental role in the descrip- n : tion of physical systems, because their many interesting properties both from mathematical v i and physical points of view. Many of these Hamiltonian systems have proved to be of an ex- X traordinaryphysicalintereste.g. harmonicoscilator, Keplerproblem,Morse[1],Posch–Teller r a [2], Smorodinski-Winternitz [3, 4], Calogero [5] and Sutherland [6] potentials. In this paper we present a class of integrable Hamiltonian systems that allow us to gener- alize the intertwining transformations for one-dimensional (1D) systems [7] to higher dimen- sions. These Hamiltonian systems are superintegrable, i.e., they have more than N constants of motion, being N the dimension of the configuration space for the Hamiltonian system. Although these motion integrals are not all of them in involution they determine more than one subset of Nconstants (in all the cases one of them is the Hamiltonian) in involution. The system is said to be superintegrable in the sense of [8] or maximally superintegrable if there exist 2N 1 invariants well defined in phase–space. − Using the Marsden–Weinstein reduction procedure[9] we construct such classical systems starting from a free Hamiltonian lying in an N-dimensional homogeneous space of a suitable 1 Lie group, whose action allows to us to calculate a momentum map that assure the inte- grability, or even superintegrability of the reduced systems [4]. From an opposite point of view, there are good reasons to suspect that any integrable system may be constructed as a reduction from a free one [10]. For the corresponding quantum systems we present here a generalization to higher dimensional spaces of the intertwining transformations for 1D factorizable systems. These 1D systems have dynamical Lie algebras of rank one generated by the intertwining operators [7]. By using a concrete superintegrable Hamiltonian system with underlying symmetry the Lie algebra su(3) we find that its dynamical symmetry can bee enlarged to so(6). However, these results can be implemented to higher dimensional systems of the same class, and also can be helpful in the study of other kinds of integrable systems using algebraic methods [11, 12, 13, 14, 15, 16]. In sections 2, 3 and 4 we will introduce a two-dimensional superintegrable system and find some separable solutions by standard procedures (Hamilton-Jacobi equation). Next, in sections 5and6westudythecorrespondingSchro¨dingerequation whichisfactorizable intwo 1D equations. We construct some sets of intertwining operators closing the Lie algebras u(3) andso(6)bytakingalsointoaccountdifferentseparablecoordinatesystemsaswellasdiscrete symmetries. We characterize theeigenfunctions of theHamiltonian hierachies, obtained from theintertwiningoperators,asirreducibleunitaryrepresentationsofthesedynamicalalgebras. 2 Superintegrable SU(p,q)-Hamiltonian systems Let us consider a free Hamiltonian H = 4gµν¯p p¯ (µ,ν = 0, ,n = p+q 1 and the bar µ ν ··· − SU(p,q) stands for complex conjugate) defined in the configuration space . This is an SU(p−1,q)×U(1) hermitian hyperbolic space with metric g and coordinates yµ C, verifying g y¯µyν = 1 µν µ¯ν ∈ (by p we denote the conjugate momenta). The geometry and properties of this kind of µ spaces are described in [17, 18]. Usingamaximalabelian subalgebra(MASA)oftheLiealgebrasu(p,q)[19]thereduction procedure allows us to obtain a reduced Hamiltonian, H = 21gµνpsµpsν + V(s), lying in the corresponding reduced space (a homogeneous SO(p,q)-space), where V(s) is a potential depending on the real coordinates sµ satisfying the constraint g sµsν = 1. µν The set of complex coordinates yµ is transformed by the reduction into a set of ignorable variables xµ (which are the parameters of the transformation associated to the MASA of u(p,q) used in the reduction) and the actual real coordinates sµ. IfY ,µ = 0, ,n,isabasisoftheconsideredMASAofu(p,q),formedbypureimaginary µ ··· matrices (this is a basic hypotesis in our reduction procedure), the relation between old (yµ) and new coordinates (xµ,sµ) is yµ = B(x)µ sν, B(x) = exp(xµY ) . ν µ Thisrelationship assurestheignorability ofthexcoordinates (thevector fieldscorresponding to the MASA are straightened out in these coordinates). The Jacobian matrix, J, of the 2 coordinate transformation is given explicitly by ∂(y,y¯) A B ∂yµ J = = , Aµ = = (Y )µ yρ. ∂(x,s) A¯ B¯ ν ∂xν ν ρ (cid:18) (cid:19) The Hamiltonian calculated in the new coordinates is written as 1 H = c gµνp p +V(s) , V(s)= pT (A†KA)−1p , 2 µ ν x x (cid:18) (cid:19) where p are the constant momenta associated to the ignorable coordinates x and K is the x matrix defined by the metric g. A detailed exposition of this construction procedure of this family of superintegrable systems can be found in [20]. 3 A classical superintegrable u(3)-Hamiltonian system To obtain the classical superintegrable Hamiltonian associated to the unitary Lie algebra su(3) using the reduction procedure sketched in the previous section we are going to proceed in the following way [21]. Letusconsiderthebasisofsu(3)determinedby3 3matrices X , ,X ,whoseexplicit 1 8 × ··· form, when using the metric K = diag(1,1,1), is i 0 0 0 0 0 0 1 0 0 i 0 X = 0 i 0 X = 0 i 0 X = 1 0 0 X = i 0 0 1 2 3 4  −    −    0 0 0 0 0 i 0 0 0 0 0 0 −         0 0 1 0 0 i 0 0 0 0 0 0 X = 0 0 0 X = 0 0 0 X = 0 0 1 X = 0 0 i . 5 6 7 8         1 0 0 i 0 0 0 1 0 0 i 0 − −         In the compact case, as here with su(3), there is only one MASA. This is the Cartan subalgebra, generated by the matrices diag(i, i,0) and diag(0,i, i). However, we shall − − work, in order to facilitate the computations, with the algebra u(3) instead of su(3). Hence, we shall use the following basis for the corresponding MASA in u(3) Y = diag(i,0,0), Y = diag(0,i,0), Y = diag(0,0,i). 0 1 2 Theactualrealcoordinatessarerelatedtothecomplexcoordinatesybyy = s eixµ (µ = µ µ 0,1,2). The Hamiltonian can be written as 1 m2 m2 m2 H = p2+p2+p2 +V(s), V(s)= 0 + 1 + 2 , (3.1) 2 0 1 2 s2 s2 s2 0 1 2 (cid:0) (cid:1) lying in the 2-sphere (s )2+(s )2+(s )2 = 1. 0 1 2 To see that the system, so obtained, is superintegrable it is necessary to construct its invariants of motion. In this case we obtain three invariants 2 s s R = (s p s p )2+ m ν +m µ , µ < ν, µ = 0,1, ν = 1,2. (3.2) µν µ ν ν µ µ ν − s s (cid:18) µ ν (cid:19) 3 Note that only two of them are in involution at the same time (being one of them the Hamiltonian), sothesystem issuperintegrablein thesenseof[8]. Thesumof theseinvariants is the Hamiltonian up to an additive constant. 4 The Hamilton-Jacobi equation for the u(3)-system Thesolutionsofthemotionproblemforthissystemcanbeobtainedsolvingthecorresponding Hamilton–Jacobi(HJ)equationinanappropriatecoordinatesystem,suchthattheHamilton– Jacobi equation separates into a system of ordinary differential equations. The 2-sphere can be parametrized on spherical coordinates (φ ,φ ) around the s axis 1 2 2 [18, 22] by s = cosφ cosφ , s = cosφ sinφ , s = sinφ (4.1) 0 2 1 1 2 1 2 2 where φ [0,2π) and φ [π/2,3π/2]. Then, the Hamiltonian (3.1) is written as 1 2 ∈ ∈ 1 p2 H = p2 + φ1 +V(φ ,φ ), 2 φ2 cos2φ 1 2 2! 1 m2 m2 m2 V(φ ,φ )= 0 + 1 + 2 . 1 2 cos2φ cos2φ sin2φ sin2φ 2 (cid:18) 1 1(cid:19) 2 The potential is periodic and has singularities along the coordinate lines φ = 0,π/2,π,3π/2, 1 and φ = π/2,3π/2, and thre is a unique minimum inside each regularity domain. 2 The invariants (3.2) (denoted by Iˆ) can be written in terms of the basis X , ,X as 1 8 { ··· } Iˆ = X2+X2,Iˆ = X2+X2 and Iˆ = X2+X2. The u(3)-Casimir is 1 3 4 2 5 6 3 7 8 C = 4X2 +2 X ,X +4X2 +3Iˆ +3Iˆ +3Iˆ , 1 { 1 2} 2 1 2 3 where the three first terms are the second order operators in the enveloping algebra of the compact Cartan subalgebra of u(3). The Hamiltonian is rewritten as H = I +I +I +constant, where by I we denote the 1 2 3 i invariant Iˆ but rewritten in spherical coordinates. So, we have i 1 m2 m 2 I = p2 + 0 + 1 , 1 2 φ1 cos2φ sin2φ 1 1 1 m2 1 m2 I = tan2φ p2 sin2φ + 0 +cos2φ p2 + 2 2 2 2 φ1 1 cos2φ 1 2 φ2 tan2φ (cid:18) 1(cid:19) (cid:18) 2(cid:19) 1 + p p sin2φ tanφ , 2 φ1 φ2 1 2 1 m2 1 m2 I = tan2φ p2 cos2φ + 1 +sin2φ p2 + 2 3 2 2 φ1 1 sin2φ 1 2 φ2 tan2φ (cid:18) 1(cid:19) (cid:18) 2(cid:19) 1 p p sin2φ tanφ . − 2 φ1 φ2 1 2 4 Now, the HJ equation takes the form 1 ∂S 2 m2 1 1 ∂S 2 m2 m2 + 2 + + 0 + 1 = E. 2(cid:18)∂φ2(cid:19) sin2φ2 cos2φ2 2 (cid:18)∂φ1(cid:19) cos2φ1 sin2φ1! It separates into two ordinary differential equations taking into account that the solution of the HJ equation can be written as S(φ ,φ )= S (φ )+S (φ ) Et. Thus, 1 2 1 1 2 2 − 1 ∂S 2 m2 m2 1 + 0 + 1 = α , 2 ∂φ cos2φ sin2φ 1 (cid:18) 1(cid:19) 1 1 1 ∂S 2 m2 α 2 + 2 + 1 = α , 2 ∂φ sin2φ cos2φ 2 (cid:18) 2(cid:19) 2 2 where α = E and α are the separation constants (which are positive). Each one of these 2 1 two equations have the same form of those corresponding to the 1D problem [21]. The solutions of both HJ equations are easily computed and can be found as particular cases in Ref. [18]. A detailed analysis of them shows that all the orbits in a neighborhood of a critical point (center) are closed and thus, the corresponding trajectories are periodic (a direct consequence of the correspondence between extrema of the potential and critical points of the phase space). The explicit solutions, when we restrict us to the domain 0 < φ ,φ < π/2, are 1 2 1 cos2φ = b + b2 4α Ecos2√2Et , 2 2E 2 2− 1 (cid:20) q (cid:21) 1 1 b2 4α m2 1/2 cos2φ = b + 1− 1 0 (b cos2φ 2α )sin2√2α β 1 2α 1 cos2φ b2 4α E 2 2− 1 1 1 1 " 2 (cid:20) 2− 1 (cid:21) (cid:0) +2√α [(b Ecos2φ )cos2φ α ]1/2cos2√2α β , 1 2 2 2 1 1 1 − − (cid:17)i where b = α +m2 m2 and b = E+α m2. 1 1 0− 1 2 1− 2 Note that in this domain for the variables (φ ,φ ) the minimum for the potential corre- 1 2 sponds to the point (φ = arctan m /m , φ = arctan m /(m +m )). The value of the 1 1 0 2 2 0 1 potential at this point is (m +m +m )2. Hence, the energy E is bounded from below, i.e. 0 p1 2 p E (m +m +m )2. 0 1 2 ≥ As we mentioned before these results reflect essentially the (1D) su(2)-case. In fact, all systems we can construct using Cartan subalgebras can be described in a unified way as it was shown in [18, 20, 21]. 5 A quantum superintegrable u(3)-Hamiltonian system From the quantum point of view the Hamiltonian (3.1) takes the form l2 1/4 l2 1/4 l2 1/4 H = J2+J2+J2 + 0 − + 1 − + 2 − , (5.1) − 0 1 2 (s )2 (s )2 (s )2 0 1 2 (cid:0) (cid:1) 5 where (l ,l ,l ) R3, and J = ǫ s ∂ . 0 1 2 i ijk j k ∈ − Also in the quantum case, the eigenvalue problem HΦ = EΦ after substituting the coordinates (4.1), takes the form of a separable differential equation l2 1/4 1 l2 1/4 l2 1/4 ∂2 +tan(φ )∂ + 2− + ∂2 + 0− + 1− Φ = EΦ. (5.2) − φ2 2 φ2 sin2(φ ) cos2(φ ) − φ1 cos2(φ ) sin2(φ ) (cid:20) 2 2 (cid:18) 1 1 (cid:19)(cid:21) Taking the solutions separated in the variables φ and φ as Φ(φ ,φ ) = f(φ )g(φ ) after 1 2 1 2 1 2 replacing in (5.2) we get the equations l2 1/4 l2 1/4 ∂2 + 0 − + 1 − f(φ ) = αf(φ ), (5.3) − φ1 cos2(φ ) sin2(φ ) 1 1 (cid:20) 1 1 (cid:21) α l2 1/4 ∂2 +tan(φ )∂ + + 2 − g(φ ) = Eg(φ ), (5.4) − φ2 2 φ2 cos2(φ ) sin2(φ ) 2 2 (cid:20) 2 2 (cid:21) where α is a separating constant. These two (one-variable dependent)equations can be solved using the standard factorizations obtaining polynomial solutions. Notice that the results obtained for the first equation will match in a certain way with those of the second one giving rise to degenerate levels. 5.1 The factorization of the φ -equation 1 The 1D Hamiltonian corresponding to the equation (5.3) in the variable φ can be factorized 1 using the theory of factorizations by Infeld and Hull [7]. The second order differential operator at the l.h.s. of equation (5.3) can be written as a product of first order operators Hφ1 = A+A−+λ , (0) 0 0 0 where A± = ∂ (l +1/2) tanφ +(l +1/2) cotφ , and λ = (l +l +1)2. Also it is 0 ± φ1 − 0 1 1 1 0 0 1 possible to construct a family of operators A+,A− ,λ ,Hφ1 , m Z, where { m m m (m)} ∈ A± = ∂ (l +m+1/2) tanφ +(l +m+1/2) cotφ , (5.5) m ± φ1 − 0 1 1 1 λ = (l +l +2m+1)2, m 0 1 (l +m)2 1/4 (l +m)2 1/4 Hφ1 = ∂2 + 0 − + 1 − . (5.6) (m) − φ1 cos2φ sin2φ 1 1 Hence, we obtain a 1D Hamiltonian hierarchy (5.6), whose first element is Hφ1, satisfies (0) Hφ1 = A+ A− +λ = A− A+ +λ . (5.7) (m) m m m m−1 m−1 m−1 From it we see that the operators A± are shape invariant intertwining operators, i.e. m A−Hφ1 = Hφ1 A− , A+Hφ1 = Hφ1 A+. (5.8) m (m) (m+1) m m (m+1) (m) m 6 Formally, theoperatorsA± acting onaHamiltonian eigenfunction give anothereigenfunction m of a consecutive Hamiltonian in the hierarchy with the same eigenvalue, i.e. A− : φ1 φ1 , A+ : φ1 φ1. m Hm → Hm+1 m Hm+1 → Hm where φ1 is the eigenfunction space of Hφ1 . Hm (m) The discrete spectrum and the physical eigenstates of Hφ1 may be obtained, in principle, (0) from the fundamental states f0 (and their eigenvalues) of all the Hamiltonians of the hier- (m) archy Hφ1 . The fundamental states are determined by the equation A− f0 = 0 whose { (m)} m (m) solutions, up to a normalization constant, are f0 (φ ) = cosl0+m+1/2(φ ) sinl1+m+1/2(φ ), (5.9) (m) 1 1 1 with eigenvalues λ = (l +l +2m+1)2. m 0 1 The excited eigenfunction fm of Hφ1 can be obtained from the ground eigenstate f0 (0) (m) (m) of Hφ1 (both with the same eigenvalue) applying consecutive operators A+ (0) fm = A+A+ A+ f0 (0) 0 1 ··· m−1 (m) (5.10) = N sinl1+1/2(φ ) cosl0+1/2(φ ) P(l1,l0)[cos(2φ )], 1 1 m 1 (a,b) where P (x) are Jacobi polynomials and N a normalization constant. The spectrum of n the Hamiltonian Hφ1 (5.3) is given by (0) α= λ = (l +l +2m+1)2, m Z≥0 . (5.11) m 0 1 ∈ 5.2 The dynamical algebras associated to the φ -factorization 1 The shape invariant intertwining operators for the 1D Hamiltonian hierarchy Hφ1 deter- { (m)} mine some Lie algebras that we can characterize as follows. Let us define free-index operators A± acting inside the total space from the above m m ⊕ H A± by [23, 24] (m) A+f := 1A+f f˜ (m+1) 2 m (m+1) ∝ (m) A−f := 1A−f f˜ (5.12) (m) 2 m (m) ∝ (m+1) Af := 1(l +l +2m)f f (m) −2 0 1 (m) ∝ (m) where f (or f˜ ) denotes an eigenfunction of H . We can rewrite (5.7) as (m) (m) (m) [A,A±]= A±, [A−,A+] = 2A (5.13) ± − assuming that the action is on any f . These commutators determine a Lie algebra su(2) (m) whose Casimir element is = A+A− +A(A 1). C − It can be proved (for more details see Ref. [25]) that the eigenstates of the Hamiltonians Hφ1 (5.9) can be characterized, if l and l are positive o zero integer numbers, in terms of (m) 0 1 7 thevectorsoftheirreducibleunitaryrepresentations(IUR)ofsu(2),labeledbytheparameter j such that 2j Z≥0. Effectively, the ground states f0 of Hφ1 are characterized by ∈ (m) (m) A−f0 = 0, Af0 = [(l +l +2m)/2]f0 . (5.14) (m) (m) − 0 1 (m) Then, we identify (up to a normalization constant) f0 = j , j , j = (l +l +2m)/2, (m) | m − mi m 0 1 where js denotes the vectors of the IUR ‘j′ of su(2). , | i The excited states of Hφ1 are obtained using expression (5.10). So, the eigenstate of the (m) k-th excited level of Hφ1 is (0) fk j +k, j +k , j = (l +l +2k)/2, k = 0,1,2... (0) ≡ | k − k i k 0 1 Moreover, Hφ1 (as well as any Hφ1 ) can be expressed in terms of the su(2)-Casimir acting (0) (m) C on such representations by Hφ1 = 4( +1/4). Hence, the eigenvalue equation for any of the (0) C excited states can be written as follows Hφ1fk 4( +1/4)j +k, j +k = 4(j +k+1/2)2 j +k, j +k (0) (0) ≡ C | 0 − 0 i 0 | 0 − 0 i = (l +l +2k+1)2fk , k = 0,1,2,... 0 1 (0) However, there is an ambiguity due that different fundamental states (5.9) with values of l and l giving the same j = (l +l )/2 would lead to the same j -representation of su(2). 0 1 0 0 1 0 Adding the diagonal operator D, Df := (l l )f , to the generators of su(2) (5.12) (m) 0 1 (m) − weobtainu(2). TheeigenstatesoftheHamiltonianhierarchyarenowcompletlycharacterized by the IUR’s of u(2). However, different u(2)-IUR’s may give rise to (different) states with the same energy. On the other hand, the states of the Hamiltonian hierarchies, when l or l are not in 0 1 Z≥0, correspond to non-unitary representations of u(2), although they and their spectra are also given by formulae (5.9) and (5.11), respectively. From a classification point of view it will be interesting to construct a new Lie algebra, obviously containing the subalgebra chain su(3) u(3), such that only one of its IUR’s ⊂ characterizes all the eigenstates in the hierarchy with same energy. Inordertobuildasuchdynamicalalgebraweneedtointroduceaatwo-subindexnotation, in terms of the parameters (l ,l ), for the intertwining operators. The Hamiltonians Hφ1 0 1 (m) (5.6) will be denoted by Hφ1 , its eigenfunctions by f , and the factor operators A± in (l0,l1) (l0,l1) 0 (5.5) will be rewritten as A± . In this way, relation (5.8) can be expressed as (l0,l1) A− Hφ1 = Hφ1 A− , A+ Hφ1 = Hφ1 A+ . (l0,l1) (l0,l1) (l0+1,l1+1) (l0,l1) (l0,l1) (l0+1,l1+1) (l0,l1) (l0,l1) We can also define the free-subindex operators A±,A,D as in (5.12). The fact that each two-parameter Hamiltonian Hφ1 is invariant under the reflections (l0,l1) I :(l ,l ) ( l ,l ), I :(l ,l ) (l , l ) 0 0 1 0 1 1 0 1 0 1 → − → − 8 originates a second factorization ([27, 28, 29]) via conjugation of the operators A±,A,D, I A±I = A˜±, I AI = A˜, I DI = D˜, 0 0 0 0 0 0 IÃ±I = A˜∓, I AI = A˜, I DI = D˜. 1 1 1 1 1 1 − − The explicit form of the new operators is 1 A˜± = ∂ +(l 1/2) tanφ +(l +1/2) cotφ , A˜ = ( l +l ). (5.15) (l0,l1) ± φ1 0− 1 1 1 (l0,l1) −2 − 0 1 These operators A˜,A˜± close a second Lie algebra su(2), denoted su(2), which commutes { } with the previous su(2). Since, moreover, D and D˜ essentially coincide with A˜ and A, respectively, the new dynamical algebra is su(2) su(2) so(4). f ⊕ ≈ The action of the so(4)-generators on a Hamiltonian Hφ1 originates a 2D parameter Z f (l0,l1) so(4)-hierarchy H , m,n , fixed by the initial values (l ,l ). Each energy { l0−n+m,l1+n+m} ∈ 0 1 level of this Hamiltonian hierarchy is degenerated and the eigenstates are characterized by so(4)-representations. 5.3 The factorization of the φ -equation 2 The second equation (5.4) can be also factorized provided that the separation constant α is substituted by the eigenvalues obtained from the φ -factorization α= λ = (l +l +2m)2. 1 m 0 1 The Hamiltonian associated to this φ -equation (5.4) is 2 (l +l +2m+1)2 l2 1/4 Hφ2 = ∂2 +tan(φ )∂ + 0 1 + 2 − . (5.16) (0) − φ2 2 φ2 cos2(φ ) sin2(φ ) 2 2 It can be factorized in terms of two first-order differential operators as Hφ2 = M+M−+µ . (0) 0 0 0 This Hamiltonian Hφ2 is the first element of the Hamiltonian hierarchy Hφ2 , n Z≥0, in (0) { (n)} ∈ the φ –variable, whose elements can be written as 2 Hφ2 = M+M− +µ = M− M+ +µ , (n) n n n n−1 n−1 n−1 M± = ∂ (l +l +2(m+1)+n)tan(φ )+(l +n+1/2)cot(φ ), n ± φ2 − 0 1 2 2 2 µ = (l +l +l +2n+2m+3/2)(l +l +l +2n+2m+5/2). (5.17) n 1 0 2 2 1 0 The energy values E are given by the factorization constant µ , i.e. E = µ . The ground n n n states g0 for this hierarchy are (n) g0 (φ ) = N cos(φ )l1+l0+2m+1sin(φ )l2+n+1/2. (n) 2 2 2 The eigenfunctions gn of the initial Hamiltonian Hφ2 (5.16) can be written as (0) (0) gn (φ ) = cos(φ )l1+l0+2m+1sin(φ )l2+1/2P(l2+1/2,l1+l0+2m+1)[cos2φ ]. (5.18) (0) 2 2 2 n 2 9 The index-free operators M±, defined in a similar way as A± in (5.12), close again a Lie algebra su(2). The eigenfunctions (5.18) are square-integrable, but the su(2)-representations are IUR provided that the parameters l ,l ,l belong to Z≥0. 0 1 2 The new factorization leads to a degeneration of the energy levels indicating that the underlying dynamical symmetry could be larger than so(4). Finally joining both factorizations we obtain the square-integrable eigenfunctions separated in the variables (φ ,φ ) of the Hamiltonian (5.2) 1 2 Φ (φ ,φ ) = fm(φ )gn (φ ), m,n Z≥0, m,n 1 2 (0) 1 (0) 2 ∈ where fm(φ and gn (φ ) are given by the expressions (5.10) and (5.18), respectively. Their (0) 1 (0) 2 corresponding eigenvalues E (5.17) are degenerated for the values of m and n whose sum n,m m+n keeps constant [26] (see Figures 1 and 2). 6 Dynamical symmetries of the u(3)-Hamiltonian hierarchy The spectrum of the u(3)-Hamiltonian system (5.1) suggestes a bigger dynamical algebra of the Hamiltonian hierarchy. By introducing three sets of intertwining operators closing an algebra u(3) and using reflexion operators this algebra is enlarged to so(6). These three sets of operators are related with three set of spherical coordinates that we can take in the 2-sphere submerged in a 3D ambient space with cartesian axes s ,s ,s . Since the axes 0 1 2 { } (s ,s ,s ) play a symmetric role in the Hamiltonian (5.1), we will take their cyclic rotations 0 1 2 to get two other intertwining sets. These two set of spherical coordinates also separate the Hamiltonian (5.1). l1 l1 4 4 2 2 -4 -2 2 4 l0 -4 -2 2 4 l0 -2 -2 -4 -4 Figure 1: Plot of thetwo IUR’sso(4)-hierarchies, whereeach pointrepresents a Hamiltonian. At the left it is the integer (l = 0,l = 0) and at the right the half-odd (l = 0,l = 1) 0 1 0 1 hierarchy. The arrows stand for the intertwining operators. 10