
SIGMA 7 (2011), 113, 11 pages arXiv:1112.2333
http://dx.doi.org/10.3842/SIGMA.2011.113
Breaking PseudoRotational Symmetry through H_{+}^{2} Metric Deformation in the Eckart Potential Problem
Nehemias LeijaMartinez ^{a}, David Edwin AlvarezCastillo ^{b} and Mariana Kirchbach ^{a}
^{a)} Institute of Physics, Autonomous University of San Luis Potosi, Av. Manuel Nava 6, San Luis Potosi, S.L.P. 78290, Mexico
^{b)} H. Niewodniczanski Institute of Nuclear Physics, Radzikowskiego 152, 31342 Kraków, Poland
Received October 12, 2011, in final form December 08, 2011; Published online December 11, 2011; Misprints are corrected December 24, 2011
Abstract
The peculiarity of the Eckart potential problem
on H_{+}^{2} (the upper sheet of the twosheeted twodimensional
hyperboloid),
to preserve the (2l+1)fold degeneracy of the states
typical for the geodesic motion there, is usually explained in casting
the respective Hamiltonian in terms of the Casimir invariant of
an so(2,1) algebra, referred to as potential algebra. In general,
there are many possible
similarity transformations of the symmetry algebras of the free motions on
curved surfaces towards potential algebras, which are not all
necessarily unitary.
In the literature, a transformation of the symmetry algebra of the geodesic
motion on H_{+}^{2} towards the potential algebra of Eckart's Hamiltonian
has been constructed for the prime purpose to prove that
the Eckart interaction belongs to the class of Natanzon potentials.
We here take a different path and search for a transformation which
connects the (2l+1) dimensional representation space of the
pseudorotational so(2,1) algebra, spanned by the rankl
pseudospherical harmonics, to the representation space of equal dimension
of the potential algebra and find a transformation of the scaling type.
Our case is that in so doing one is producing a deformed isometry copy
to H_{+}^{2} such that the free motion on the copy
is equivalent to a motion on H_{+}^{2}, perturbed by a
coth interaction.
In this way, we link the so(2,1) potential algebra concept of the
Eckart Hamiltonian to a subtle type of pseudorotational symmetry breaking
through H_{+}^{2} metric deformation.
From a technical point of view, the results reported here are obtained by
virtue of certain nonlinear finite expansions of Jacobi polynomials into
pseudospherical harmonics. In due places, the pseudorotational
case is paralleled by its so(3) compact analogue, the cotangent
perturbed motion on S^{2}.
We expect awareness of different so(2,1)/so(3) isometry copies
to benefit simulation studies on curved manifolds of manybody systems.
Key words:
pseudorotational symmetry; Eckart potential; symmetry breaking through metric deformation.
pdf (467 Kb)
tex (130 Kb)
[previous version:
pdf (466 kb)
tex (130 kb)]
References
 Natanzon G.A.,
General properties of potentials for which the Schrödinger equation can be solved by means of hyper geometric functions,
Theoret. and Math. Phys. 38 (1979), 146153.
 Alhassid Y., Gürsey F., Iachello F.,
Potential scattering, transfer matrix, and group theory,
Phys. Rev. Lett. 50 (1983), 873876.
 Engelfield M.J., Quesne C.,
Dynamical potential algebras for Gendenshtein and Morse potentials,
J. Phys. A: Math. Gen. 24 (1991), 35573574.
 Manning M.F., Rosen N.,
Potential functions for vibration of diatomic molecules,
Phys. Rev. 44 (1933), 951954.
 Wu J., Alhassid Y.,
The potential group approach and hypergeometric differential equations,
J. Math. Phys. 31 (1990), 557562.
Wu J., Alhassid Y., Gürsey F.,
Group theory approach to scattering. IV. Solvable potentials associated with SO(2,2),
Ann. Physics 196 (1989), 163181.
 Levai G.,
Solvable potentials associated with su(1,1) algebras: a systematic study,
J. Phys. A: Math. Gen. 27 (1994), 38093828.
 Cordero P., Salamó S.,
Algebraic solution for the Natanzon hypergeometric potentials,
J. Math. Phys. 35 (1994), 33013307.
 Cordriansky S., Cordero P., Salamó S.,
On the generalized Morse potential,
J. Phys. A: Math. Gen. 32 (1999), 62876293.
 Gangopadhyaya A., Mallow J.V., Sukhatme U.P.,
Translational shape invariance and inherent potential algebra,
Phys. Rev. A 58 (1998), 42874292.
 Rasinariu C., Mallow J.V., Gangopadhyaya A.,
Exactly solvable problems of quantum mechanics and their spectrum generating algebras: a review,
Cent. Eur. J. Phys. 5 (2007), 111134.
 Kalnins E.G., Miller W. Jr., Pogosyan G.,
Superintegrability on the twodimensional hyperboloid,
J. Math. Phys. 38 (1997), 54165433.
Berntson B.K.,
Classical and quantum analogues of the Kepler problem in nonEuclidean geometries of constant curvature,
B.Sc. Thesis, University of Minnesota, 2011.
 Gazeau J.P.,
Coherent states in quantum physics,
WileyVCH, Weinheim, 2009.
 Bogdanova I., Vandergheynst P., Gazeau J.P.,
Continuous wavelet transformation on the hyperboloid,
Appl. Comput. Harmon. Anal. 23 (2007), 286306.
 Kim Y.S., Noz M.E.,
Theory and application of the Poincaré group,
D. Reidel Publishing Co., Dordrecht, 1986.
 De R., Dutt R., Sukhatme U.,
Mapping of shape invariant potentials under point canonical transformations,
J. Phys. A: Math. Gen. 25 (1992), L843L850.
 AlvarezCastillo D. E., Compean C.B., Kirchbach M.,
Rotational symmetry and degeneracy: a cotangent perturbed rigid rotator of unperturbed level multiplicity,
Mol. Phys. 109 (2011), 14771483,
arXiv:1105.1354.
 Raposo A., Weber H.J., AlvarezCastillo D.E., Kirchbach M.,
Romanovski polynomials in selected physics problems,
Cent. Eur. J. Phys. 5 (2007), 253284,
arXiv:0706.3897.
 Higgs P.W.,
Dynamical symmetries in a spherical geometry. I,
J. Phys. A: Math. Gen. 12 (1979), 309323.

