Breaking Pseudo-Rotational Symmetry through H 2 + Metric Deformation in the Eckart Potential Problem
Nehemias Leija-Martinez, David Edwin Alvarez-Castillo, Mariana Kirchbach
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2011), 113, 11 pages Breaking Pseudo-Rotational Symmetry through H Metric Deformation in the Eckart Potential Problem
Nehemias LEIJA-MARTINEZ † , David Edwin ALVAREZ-CASTILLO ‡ and Mariana KIRCHBACH †† Institute of Physics, Autonomous University of San Luis Potosi,Av. Manuel Nava 6, San Luis Potosi, S.L.P. 78290, Mexico
E-mail: nemy@ifisica.uaslp.mx, mariana@ifisica.uaslp.mx ‡ H. Niewodnicza´nski Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Krak´ow, Poland
E-mail: [email protected]
Received October 12, 2011, in final form December 08, 2011; Published online December 11, 2011http://dx.doi.org/10.3842/SIGMA.2011.113
Abstract.
The peculiarity of the Eckart potential problem on H (the upper sheet of thetwo-sheeted two-dimensional hyperboloid), to preserve the (2 l + 1)-fold degeneracy of thestates typical for the geodesic motion there, is usually explained in casting the respectiveHamiltonian in terms of the Casimir invariant of an so(2,1) algebra, referred to as potentialalgebra. In general, there are many possible similarity transformations of the symmetryalgebras of the free motions on curved surfaces towards potential algebras, which are notall necessarily unitary. In the literature, a transformation of the symmetry algebra of thegeodesic motion on H towards the potential algebra of Eckart’s Hamiltonian has beenconstructed for the prime purpose to prove that the Eckart interaction belongs to the classof Natanzon potentials. We here take a different path and search for a transformationwhich connects the (2 l +1) dimensional representation space of the pseudo-rotational so(2,1)algebra, spanned by the rank- l pseudo-spherical harmonics, to the representation space ofequal dimension of the potential algebra and find a transformation of the scaling type. Ourcase is that in so doing one is producing a deformed isometry copy to H such that thefree motion on the copy is equivalent to a motion on H , perturbed by a coth interaction.In this way, we link the so(2,1) potential algebra concept of the Eckart Hamiltonian toa subtle type of pseudo-rotational symmetry breaking through H metric deformation.From a technical point of view, the results reported here are obtained by virtue of certainnonlinear finite expansions of Jacobi polynomials into pseudo-spherical harmonics. In dueplaces, the pseudo-rotational case is paralleled by its so(3) compact analogue, the cotangentperturbed motion on S . We expect awareness of different so(2,1)/so(3) isometry copies tobenefit simulation studies on curved manifolds of many-body systems. Key words: pseudo-rotational symmetry; Eckart potential; symmetry breaking through met-ric deformation
Group theoretical approaches to bound state problems in physics have played a pivotal r´ole inour understanding of spectra classifications. It is a well known fact, that several of the exactlysolvable quantum mechanical potentials give rise to spectra which fall into the irreducible repre-sentations of certain Lie groups. As a representative case, we wish to mention the widely studiedclass of Natanzon potentials [1] known to produce spectra that populate multiplets of the pseudo-rotational algebras so(2,2)/so(2,1). This phenomenon has been well understood in casting theHamiltonians of the potentials under consideration as Casimir invariants of so(2,2)/so(2,1) alge-bras in representations not necessarily unitarily equivalent to the pseudo-rotational and referred a r X i v : . [ m a t h - ph ] D ec N. Leija-Martinez, D.E. Alvarez-Castillo and M. Kirchbachto as “potential algebras” [2, 3]. One popular representative of the Natanzon class potentialsis the Eckart potential, suggested by Manning and Rosen [4] for the description of the vibra-tional modes of diatomic molecules. Its so(2,2)/so(2,1) potential algebras have been constructed,among others, in [5, 6, 7, 8, 9, 10]. One can start with considering the Eckart potential on H ,the two-sheeted 2D hyperbolic surface, defined as H ± : z − x − y = R , where R is a constant. This geometry has been studied in great detail [11], and finds applications,among others, as a coherent state manifold [12, 13]. One usually chooses the upper sheet H ,which corresponds to the following parametrization in global coordinates z = R cosh η, x = R sinh η cos ϕ,y = R sinh η sin ϕ, z > (cid:112) x + y , η ∈ [0 , ∞ ) . The free geodesic motion on H is now determined by the eigenvalue problem of the squaredpseudo-angular momentum operator CHY ml ( η, ϕ ) = − (cid:126) M l ( l + 1) Y ml ( η, ϕ ) , H = − (cid:126) M C , C = 1sinh η ∂∂η sinh η ∂∂η + 1sinh η ∂ ∂ϕ , (1)so that the symmetry of the Hamiltonian, H , is so(2,1). Here, Y ml ( η, ϕ ) are the standard pseudo-spherical harmonics [14] Y ml ( η, ϕ ) = P ml (cosh η ) e imϕ , η ∈ ( −∞ , + ∞ ) , (2)with P ml (cosh η ) being the associated Legendre functions of a hyperbolic cosines argument. In [9]it has been shown that the following particular similarity transformation of C F ( η ) C F − ( η ) = (cid:101) C , F ( η ) = (cid:115) sinh 2 g ( r ) g (cid:48) ( r ) , tanh g ( r ) = z = e − r , (3)transforms equation (1) into a Natanzon equation in the z variable, and to the central Eckartpotential problem in ordinary 3D flat position space in the r variable. In this fashion it becamepossible to cast the Schr¨odinger Hamilton operator with the Eckart potential, treated as a centralinteraction, in the form of a particular Casimir invariant of the so(2,1) algebra.Compared to [5], and [9], we here exclusively focus on the relationship between the free, andthe coth η perturbed motions on H and find a transformation that connects the ( C + 2 b coth η )-and C -eigenvalue problems and their respective invariant spaces, a transformation which has notbeen reported in the literature so far. The C invariant spaces are (2 l + 1)-dimensional, non-unitary, and spanned by the pseudo-spherical harmonics in (2). Towards our goal, we developa technique for solving the ( C + 2 b coth η )-eigenvalue problem which slightly differs from the oneof standard use. Ordinarily, the respective 1D Schr¨odinger equation is resolved by reducing it tothe hyper-geometric differential equation satisfied by the Jacobi polynomials. We here insteadconstruct the solutions directly in the basis of the free geodesic motion, i.e. in the Y ml ( η, ϕ )basis. In so doing, we find superpositions of exponentially damped pseudo-spherical harmonicswhich describe the perturbance by a hyperbolic cotangent function of the free geodesic motionon H . The matrix similarity transformation between the C and ( C + 2 b coth η ) eigenvalueproblems is then read off from those decompositions which allow to interpret the Hamilto-nian of the perturbed motion as a Casimir invariant of an so(2,1) algebra in a representationreaking Pseudo-Rotational Symmetry 3unitarily nonequivalent to the pseudo-rotational. The consequence is a breakdown of the pseudo-rotational invariance of the free geodesic motion only at the level of the representation functionswhich becomes apparent through a deformation of the metric of the hyperbolic space, and with-out a breakdown of the (2 l + 1)-fold degeneracy patterns in the spectrum. Moreover, due tocancellation effects occurring by virtue of specific recurrence relations among associated Le-gendre functions, the similarity transformation presents itself simple and, as explained in theconcluding summary section, may specifically benefit applications in problems with an appro-ximate so(2,1) symmetry. The problem under investigation is well known to transform into theRosen–Morse potential problem on S by a complexification of the hyperbolic angle, followed bya complexification of the strength of the coth η potential. On this basis, the statement on thesymmetry and degeneracy properties of the Eckart potential on H can easily be transferred tothe trigonometric Rosen–Morse potential problem on S .The paper is structured as follows. In the next section we present the eigenvalue problemof the coth-perturbed particle motion on H . There, we furthermore work out the respectivesolutions as expansions in the basis of the standard pseudo-spherical harmonics, find the non-unitary transformation relating the free and the perturbed geodesic motions, present the classof recurrence relations among associated Legendre functions that triggers the above similaritytransformation. In due places, the conclusions regarding the Rosen–Morse potential problemon S have been drawn in parallel to those regarding the Eckart potential. The paper ends withconcise summary and conclusions. and S and representationsfor the so(2,1) and so(3) algebras unitarily nonequivalentto the respective pseudo-rotational and rotational ones We begin with taking a closer look on equation (1) for the particular case of l = 0 − (cid:126) M CY ( η, ϕ ) = 0 , (4)and subject it to a similarity transformation to obtain − (cid:126) M (cid:2) F − ( η ) C F ( η ) (cid:3) (cid:2) F − ( η ) Y ( η, ϕ ) (cid:3) = − (cid:126) M (cid:101) C (cid:2) F − ( η ) Y ( η, ϕ ) (cid:3) = 0 , (5)with (cid:101) C = F − ( η ) C F ( η ). The general expression for any transformation in (5) is easily workedout (cf. [9]) and reads (cid:101) C = F − ( η ) C F ( η ) = C + 1 F ( η ) d F ( η )d η + 2 1 F ( η ) d F ( η )d η (cid:18) dd η + 12 coth η (cid:19) . We here chose F ( η ) differently than in (3) and consider the following scaling similarity trans-formation F ( η ) = e α ( l =0) η , α ( l =0) = 4 b. (6)Substitution of (6) into (5) amounts to − (cid:126) M (cid:101) C (cid:101) Y ( η, ϕ ) = − (cid:126) M (cid:104) e − α ( l =0) η C e α ( l =0) η (cid:105) (cid:104) e − α ( l =0) η Y ( η, ϕ ) (cid:105) N. Leija-Martinez, D.E. Alvarez-Castillo and M. Kirchbach
Figure 1.
Breaking of the pseudo-rotational symmetry at the level of the metric. A regular H hyperboloid associated with the so(1,2) scalar, ( z − x − y ) (left), and its non-unitary deformation,exp( − α ( l =0) η/ z − x − y ) (right) with α ( l =0) = 4 b , and η = coth − z √ x + y . Schematic presentationfor b = 1. = − (cid:126) M (cid:32) C − α l =0) b coth η (cid:33) (cid:101) Y ( η, ϕ ) = 0 , (cid:101) Y ( η, ϕ ) = e − α ( l =0) η Y ( η, ϕ ) , (7)which turns equivalent to the standard form of the coth η perturbed motion on H , for theground state − (cid:126) M ( C + 2 b coth η ) (cid:101) Y ( η, ϕ ) = − (cid:126) M b (cid:101) Y ( η, ϕ ) , (8)where the special notation, (cid:101) Y ( η, ϕ ), has been introduced in (7) for the exponentially dampedpseudo-spherical harmonic under consideration. We now notice that C in (4) is the Casimirinvariant of the so(1,2) isometry algebra of the H hyperboloid, described by the |Y ( η, ϕ ) | unitsurface, displayed in the l.h.s. of Fig. 1, and that, correspondingly, (cid:101) C = e − α ( l =0) η C e α ( l =0) η in (7)is same on the deformed surface e − α ( l =0) η/ |Y ( η, ϕ ) | , shown on the r.h.s. in Fig. 1.From this perspective, equation (7) shows that the free motion on the deformed metric inFig. 1 is equivalent to the coth η perturbed motion on H . The rest of the article is devoted togeneralize equation (7) to higher l values. From now onwards we shall switch to dimensionlessunits, (cid:126) = 1, 2 M = 1, for the sake of simplicity.In parallel, we notice that as long as so(1,2), and so(3) are related by a Wigner rotation, C in equation (1) is related to the well known expression of the standard squared orbital angularmomentum L L = − θ ∂∂θ sin θ ∂∂θ − θ ∂ ∂ϕ , by a complexification of the polar angle L θ → iη −→ C . It is the type of complexification that takes the hyperboloid H to the sphere S . Correspon-dingly, the scaling transformation in equation (6) will take the free geodesic motion on S to theone perturbed by the trigonometric Rosen–Morse potential there (cid:101) L (cid:101) Y ( θ, ϕ ) = (cid:32) L − b cot θ − α l =0) (cid:33) (cid:101) Y ( θ, ϕ ) = 0 , (cid:101) Y ( θ, ϕ ) = e − α ( l =0) θ Y ( θ, ϕ ) . (9)The counterpart of Fig. 1 is displayed on Fig. 2.reaking Pseudo-Rotational Symmetry 5 Figure 2.
The S spherical metric, | Y ( θ, ϕ ) | (left) in comparison to the exponentially deformed one, | (cid:101) Y ( θ, ϕ ) | (right) for b=1. The cotangent perturbed rigid rotor on S is equivalent (up to a shift byconstant) to free motion on the deformed metric, as visible from equation (9). We now consider the general case of a perturbance of the free geodesic motion in (1) by a hy-perbolic cotangent potential[ C + 2 b coth η ] X ( η, ϕ )= (cid:20) η ∂∂η sinh η ∂∂η + 1sinh η ∂ ∂ϕ + 2 b coth η (cid:21) X ( η, ϕ ) = − (cid:15) X ( η, ϕ ) , (10) X ( η, ϕ ) = e − αη F ( η ) e i (cid:101) mϕ , (11)where (cid:15) stands for the energy in dimensionless units. The η and ϕ dependencies of the eigen-functions separate, and we admitted for the possibility that (cid:101) m may not take same value as themagnetic quantum number in the free case. In what follows we attach indices to X ( η, ϕ ) as X ( η, ϕ ) −→ X (cid:101) ml ( η, ϕ ), and search for ( C + 2 b coth η ) eigenfunctions in the form of superpositionsof damped pseudo-spherical harmonics, (cid:101) Y ml ( η, ϕ ), with m ∈ [ (cid:101) m, l ], according to X (cid:101) ml ( η, ϕ ) = (cid:34) m = l (cid:88) m = (cid:101) m a l ( (cid:101) m +1)( m +1) e − imϕ (cid:101) Y ml ( η, ϕ ) (cid:35) e i (cid:101) mϕ , (cid:101) Y ml ( η, ϕ ) = e − αη Y ml ( η, ϕ ) . (12)Furthermore, a l ( (cid:101) m +1)( m +1) e i ( (cid:101) m − m ) ϕ (with constant a l ( (cid:101) m +1)( m +1) ) are elements of an ( l +1) × ( l +1)dimensional matrix. We begin with the observation that upon substituting (12) in (10), theparticle motion on H , hindered by a coth η interaction, equivalently rewrites to e − αη (cid:20) η ∂∂η sinh η ∂∂η − (cid:101) m sinh η + α αD (cid:21) l (cid:88) | m | = | (cid:101) m | a l ( | (cid:101) m | +1)( | m | +1) P | m | l (cosh η ) e i (cid:101) mϕ = − (cid:15)e − αη l (cid:88) | m | = | (cid:101) m | a l ( | (cid:101) m | +1)( | m | +1) P | m | l (cosh η ) e i (cid:101) mϕ , D = (cid:18) bα − (cid:19) coth η − ∂∂η , (13)where we dragged the exponential function through C from the very right to the very left, andinserted equation (12) for X (cid:101) ml ( η, ϕ ). Next we consider the particular case when the sum inequation (12) contains one term only, i.e. when (cid:101) m = l : X ll ( η, ϕ ) = a l ( l +1) , ( l +1) e − αη P ll (cosh η ) e ilϕ . N. Leija-Martinez, D.E. Alvarez-Castillo and M. KirchbachIn this case, the part of (13) behind the exponential would reduce to the differential equationfor the associated Legendre functions, shifted by the constant, α /
4, provided DP ll (cosh η ) = 0 , P ll (cosh η ) = sinh l η, (14)were to hold valid. The latter equation imposes the following condition on αα −→ α l , α l = 2 bl + , (15)where we re-labeled α to α → α l . In effect, the ( C + 2 b coth η ) eigenvalue problem simplifies tothe C eigenvalue problem subjected to a scaling similarity transformation according to[ C + 2 b coth η ] X ll ( η, ϕ ) = (cid:20) e − αlη (cid:18) C + α l (cid:19) e αlη (cid:21) X ll ( η, ϕ ) = (cid:18) l ( l + 1) + α l (cid:19) X ll ( η, ϕ ) , X ll ( η, ϕ ) = a l ( l +1) , ( l +1) e − αlη Y ll ( η, ϕ ) . (16)We set a l ( l +1) , ( l +1) = 1 for simplicity. As a next step, we consider the case of a two-termdecomposition in (12), when X l − l ( η, ϕ ) = e − αlη (cid:16) a lll P l − l (cosh η ) + a l ( l )( l +1) P ll (cosh η ) (cid:17) e i ( l − ϕ . Substituting in (10), and dragging the exponential factor again from the very right to the veryleft, amounts to[ C + 2 b coth η ] X l − l ( η, ϕ ) = e − αlη (cid:20) η ∂∂η sinh η ∂∂η − ( l − sinh η + α l α l D l (cid:21) × (cid:16) a lll P l − l (cosh η ) + a l ( l )( l +1) P ll (cosh η ) (cid:17) e i ( l − ϕ = (cid:18) l ( l + 1) + α l (cid:19) e − αη (cid:16) a lll P l − l (cosh η ) + a l ( l )( l +1) P ll (cosh η ) (cid:17) e i ( l − ϕ ,D l = (cid:18) l coth η − ∂∂η (cid:19) , (17)where the expression for D l , corresponding to D from (13) with an attached label (cid:96) , i.e. D → D l ,has been obtained in making use use of (15). The unknown constant, a l ( l )( l +1) , is now determinedfrom the condition − ( l − sinh η a l ( l )( l +1) P ll (cosh η ) + α l D l a lll P l − l (cosh η ) = − l sinh η a l ( l )( l +1) P ll (cosh η ) , setting once again a lll = 1. Taking into account that P l − l (cosh η ) = sinh l − η cosh η , one arrivesat the constraint a l ( l )( l +1) = α l − l + ( l − , thus fixing the a ll ( l +1) value. In consequence, the eigenvalue problem in equation (17) simplifiesto same form as previously found in equation (16), namely[ C + 2 b coth η ] X l − l ( η, ϕ ) = (cid:20) e − αlη (cid:18) C + α l (cid:19) e αlη (cid:21) X l − l ( η, ϕ )reaking Pseudo-Rotational Symmetry 7= (cid:34) (cid:101) C + α l ) (cid:35) X l − l ( η, ϕ ) = (cid:32) l ( l + 1) + α l ) (cid:33) X l − l ( η, ϕ ) . Proceeding successively in this way, the coefficients a l ( (cid:101) m +1)( m +1) can be found imposing thefollowing condition (cid:20) − (cid:101) m sinh η + α l D l (cid:21) (cid:34) m = l (cid:88) m = (cid:101) m a l ( (cid:101) m +1)( m +1) e − imϕ Y ml ( η, ϕ ) (cid:35) = − (cid:34) m = l (cid:88) m = (cid:101) m a l ( (cid:101) m +1)( m +1) e − imϕ m sinh η Y ml ( η, ϕ ) (cid:35) , (18)guaranteed by virtue of recurrence relations among associated Legendre functions of the type D ( l =1) P (cosh η ) = 1 sinh η P (cosh η ) , (19) D ( l =2) P (cosh η ) = 1sinh η P (cosh η ) , (20) D ( l =2) P (cosh η ) = 23 1sinh η P (cosh η ) , etc. (21)In combination with the identity, P | m | l (cosh η ) = e − imϕ Y ml ( η, ϕ ), equations (18)–(21) allow tocast the general ( C + 2 b coth η )) X (cid:101) ml ( η, ϕ ) eigenvalue problem for any l and (cid:101) m into the followingequivalent form[ C + 2 b coth η ] X (cid:101) ml ( η, ϕ ) = (cid:20) (cid:101) C + α l (cid:21) X (cid:101) ml ( η, ϕ ) = (cid:18) l ( l + 1) + α l (cid:19) X (cid:101) ml ( η, ϕ ) , (22)with X (cid:101) ml ( η, ϕ ) standing for the exact solutions of the Eckart potential on H obtained fromthe ansatz in (12). Omitting normalization factors for simplicity, the eigenfunctions to (22) forsome of the lowest l values, now cast in matrix form, are calculated as (cid:18) X ( η, ϕ ) X ( η, ϕ ) (cid:19) = e − bη (cid:18) − b e − iϕ (cid:19) (cid:18) Y ( η, ϕ ) Y ( η, ϕ ) (cid:19) , (23) X ( η, ϕ ) X ( η, ϕ ) X ( η, ϕ ) = e − bη − b e − iϕ b e − iϕ − b e − iϕ Y ( η, ϕ ) Y ( η, ϕ ) Y ( η, ϕ ) . (24)Admittedly, the above expressions are easier obtained parting from the (unnormalized) solutionsof equation (10) known from the literature [15] to be expressed in terms of Jacobi polynomials,here denoted by P γ,δn , as X (cid:101) ml ( η, ϕ ) = sinh l ηe − αlη P γ l ,δ l n (coth η ) e i (cid:101) mϕ ,γ l = bl + − (cid:18) l + 12 (cid:19) , δ l = − bl + − (cid:18) l + 12 (cid:19) , l = (cid:101) m + n. (25)Comparison of (25) to (12) allows to conclude on the existence of finite nonlinear decompositionsof the Jacobi polynomials into associated Legendre functions ( P ml ) according tosinh l η P γ l ,δ l n (coth η ) = m = l (cid:88) m = (cid:101) m a l ( (cid:101) m +1)( m +1) P ml (cosh η ) . N. Leija-Martinez, D.E. Alvarez-Castillo and M. KirchbachThe latter equation can equally well be used to pin down the expansion coefficients upon usingthe orthogonality properties of the associated Legendre functions.The ( C + 2 b coth η ) eigenvalue problem is closely related to the rigid rotator problem on S perturbed by a cotangent interaction, ( L − b cot θ ). The two problems have several features incommon. Also the cot θ interaction preserves the (2 l +1)-fold degeneracy patterns characterizingthe spectrum of the free geodesic motion, despite its non-commutativity with L . The similaritiesbetween these two cases are due to their interrelation by the following complexifications C + 2 b coth η η → iθ,b →− ib −→ L − b cot θ. (26)It is straightforward to prove that in effect of the complexifications in (26), the equation (22)is transformed to (cid:2) L − b cot θ (cid:3) X (cid:101) ml ( θ, ϕ ) = (cid:20) e − αlη (cid:18) L − α l (cid:19) e αlη (cid:21) X (cid:101) ml ( θ, ϕ )= (cid:18)(cid:101) L − α l (cid:19) X (cid:101) ml ( θ, ϕ ) = (cid:32) l ( l + 1) − α l ) (cid:33) X (cid:101) ml ( θ, ϕ ) , (27)with X (cid:101) ml ( θ, ϕ ) being given as X (cid:101) ml ( θ, ϕ ) = (cid:34) m = l (cid:88) m = (cid:101) m c l a l ( (cid:101) m +1)( m +1) e − imϕ (cid:101) Y ml ( θ, ϕ ) (cid:35) e i (cid:101) mϕ , (cid:101) Y ml ( θ, ϕ ) = e − αlθ Y ml ( θ, ϕ ) , (28)where Y ml ( θ, ϕ ) are the standard spherical harmonics, and the constants | c l | = 1 account forpossible sign changes in a l ( (cid:101) m +1)( m +1) in depending on the power of ( ib ) contained there. Inconclusion, also the cotangent perturbed rigid rotator can be cast into the form of a Casimirinvariant of the so(3) algebra though in a representation unitarily nonequivalent to the rotational.Also for the latter case the exact solutions of equation (27) are known [16], and expressed interms of real Romanovski polynomials [17] as X (cid:101) ml ( θ, ϕ ) = e − αθ/ sin l θR bl + 12 , − ( l + ) n (cot θ ) e i (cid:101) mϕ , (29)and also here in comparing (28) to (29) one finds finite nonlinear decompositions of Romanovskipolynomials into spherical harmonics according tosin l θR bl + 12 , − ( l + ) n (cot θ ) = m = l (cid:88) m = (cid:101) m c l a l ( (cid:101) m +1)( m +1) P ml ( θ ) . The Romanovski polynomials satisfy the following differential hyper-geometric equation (cid:0) x (cid:1) d R α,βn d x + 2 (cid:16) α βx (cid:17) d R α,βn d x − n (2 β + n − R α,βn = 0 . They are obtained from the following weight function ω α,β ( x ) = (cid:0) x (cid:1) β − exp (cid:0) − α cot − x (cid:1) , by means of the Rodrigues formula R α,βn ( x ) = 1 ω α,β ( x ) d n d x n (cid:104)(cid:0) x (cid:1) n ω α,β ( x ) (cid:105) . reaking Pseudo-Rotational Symmetry 9Although they are related to the Jacobi polynomials as R α,βn ( x ) = i n P − β − i α , − β + i α n ( ix ) , (30)within Bochner’s classification scheme they appear as one of the five independent polynomialsolutions of the hyper-geometric differential equation. Within this context, equation (30) doesnot rule out the necessity of considering the Romanovski polynomials, rather, it presents itselfas one out of many possible interrelationships among polynomials, valid only under certainrestrictions of the parameters of at least one of the involved polynomials.Another known example of such an interrelationship is provided by the possibility of estab-lishing the following link between (unnormalized) associated Legendre functions P ml and Jacobipolynomials P ml (cos θ ) = sin m θ P m,ml − m (cos θ ) . Obviously, the latter equation does not rule out the associated Legendre functions as a mathe-matical entity on its own in favor of Jacobi polynomials with parameters restricted in this veryparticular way.Back to the ( L − b cot θ ) eigenvalue problem, it is of interest in the spectroscopy of diatomicmolecules and has been investigated in the work [16] prior to this, where one can find explicitexpressions of the expansion coefficients for some of the lowest l values. However, there thestudy has been carried out from a predominantly spectroscopic, and significantly subordinatealgebraic perspective. Compared to [16], the present work is entirely focused on the algebraicaspect of the particle motion on the curved surface of interest, which parallels the formationmechanism of the non-unitary similarity transformation connecting C , and ( C + 2 b coth η ). Inaddition, we wish to emphasize, that casting the cotangent perturbed motion on S in (27) asa Casimir invariant of an intact geometric so(3) algebra, provides a simple alternative to thedeformed dynamic so(3) Higgs algebra [18], which approaches same problem from the perspectiveof a rotational generator on the plane tangential to the North pole of the sphere, complementedby the two components of a properly designed Runge–Lenz vector there. Equations (23), (24) can be generalized to arbitrary l according to X l ( η, ϕ ) = X l ( η, ϕ ) X l ( η, ϕ ) . . . X ll ( η, ϕ ) = e − αlη F l ( η ) e i ϕ F l ( η ) e iϕ · · · F ll ( η ) e ilϕ = A l ( ϕ ) e − αlη Y l ( η, ϕ ) Y l ( η, ϕ ) . . . Y ll ( η, ϕ ) ,A l ( ϕ ) = a l a l e − iϕ . . . a l l +1) e − ilϕ a l ... a l l +1) e − i ( l − ϕ . . . . . . . . . . . . a l ( l +1)( l +1) . (31)Then the matrix form of equation (22), after accounting for (11), extends correspondingly to (cid:20) e − αlη A l ( ϕ ) C e αlη A − l ( ϕ ) + α l (cid:21) X l ( η, ϕ ) , = − (cid:15) l X l ( η, ϕ ) ,(cid:15) l = − l ( l + 1) − α l − l ( l + 1) − b (cid:0) l + (cid:1) . (32)0 N. Leija-Martinez, D.E. Alvarez-Castillo and M. KirchbachIn this way, the eigenvalue problem in equation (10) takes its final form( C + 2 b coth η ) X l ( η, ϕ ) = (cid:20) (cid:101) C + α l (cid:21) X l ( η, ϕ ) , (cid:101) C = e − αtη A l ( ϕ ) C e αtη A − l ( ϕ ) , (33)with C being extended to a matrix encoding the dimensionality of the so(2,1) representationspace under consideration. The explicit matrices A l ( ϕ ) for l = 1 and l = 2 were given inequations (23), (24). As a reminder, equation (33) can be obtained from equation (13) in combi-nation with the recurrence relations in equations (18)–(21). Equation (33) defines a particularrepresentation of the so(2,1) algebra whose eigenvalue problem is related to the standard pseudo-rotational one by a dilation transformation. The (cid:101) C operator realization is unitarily nonequivalentto C and the motion of a particle on H , perturbed by the coth η potential, breaks the pseudo-rotational invariance at the level of the representation functions. However, the degeneracy isdefined by the eigenvalues of the Casimir invariant of the algebra and does not depend on theparticular realization of the algebra, a reason for which the perturbed spectrum in (32) carriessame degeneracy patterns as the unperturbed one, corresponding to b = 0.Finally, obtaining the counterpart to equation (33) for the case of the cotangent perturbedrigid rotator on S along the line of equations (26)–(28) is too simple an exercise as to beexplicitly worked out here, and we omit it in favor of a keeping the presentation more concise. In the present study, attention was drawn to relevance for physics problems of non-unitarysimilarity transformations of some isometry algebras of curved surfaces. Specifically, the so(1,2)isometry algebra of the two-dimensional two-sheeted hyperboloid was considered in detail anda non-unitary similarity transformation was found that connected the eigenvalue problems of theCasimir operator of the free geodesic motion and the one perturbed by a hyperbolic cotangentinteraction.The similarity transformation was concluded from transparent finite decompositions of theexact solutions of the Eckart potential in the bases of exponentially scaled pseudo-sphericalharmonics presented in the above equations (24), and (31). It furthermore was motivated bythe relevance of specific recurrence relations (19)–(21) among associated Legendre functions.A merit of representing the exact solutions of the Eckart potential as superpositions of dampedpseudo-spherical harmonics is that these expansions would provide a practical tool in the de-scription of quantum mechanical systems on H whose interactions are only approximatelydescribed by the Eckart potential, in which case the elements of the matrices A l ( ϕ ) could beconsidered as parameters to be adjusted to data.The perturbed geodesic motion on H in terms of the Eckart potential as considered here, hadthe peculiarity that the perturbance respected the degeneracy of the unperturbed free geodesicmotion though it broke the isometry group symmetry at the level of the representation functions.The case under consideration easily extended to the cotangent perturbed rigid rotator on S andverified similar observations earlier reported in [16]. In this fashion, two examples of a symmetrybreaking at the level of the representation functions have been constructed, breakdowns thatappeared camouflaged by the conservation of the degeneracy patterns in the spectra.This subtle type of symmetry breaking was visualized in Figs. 1 and 2 through the deformationof the metric of the respective H hyperboloid, and the S sphere, by the exponential scalings e − αη/ |Y ( η, ϕ ), and e − αθ/ |Y ( θ, ϕ ).reaking Pseudo-Rotational Symmetry 11 Acknowledgments
We thank Jose Limon Castillo for constant assistance in managing computer matters. Workpartly supported by CONACyT-M´exico under grant number CB-2006-01/61286.
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