Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 113, 11 pages      arXiv:1112.2333      https://doi.org/10.3842/SIGMA.2011.113

Breaking Pseudo-Rotational Symmetry through H+2 Metric Deformation in the Eckart Potential Problem

Nehemias Leija-Martinez a, David Edwin Alvarez-Castillo 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, 31-342 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 two-sheeted two-dimensional 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 pseudo-rotational so(2,1) algebra, spanned by the rank-l pseudo-spherical 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 pseudo-rotational 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 pseudo-spherical harmonics. In due places, the pseudo-rotational case is paralleled by its so(3) compact analogue, the cotangent perturbed motion on S2. We expect awareness of different so(2,1)/so(3) isometry copies to benefit simulation studies on curved manifolds of many-body systems.

Key words: pseudo-rotational symmetry; Eckart potential; symmetry breaking through metric deformation.

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References

  1. 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), 146-153.
  2. Alhassid Y., Gürsey F., Iachello F., Potential scattering, transfer matrix, and group theory, Phys. Rev. Lett. 50 (1983), 873-876.
  3. Engelfield M.J., Quesne C., Dynamical potential algebras for Gendenshtein and Morse potentials, J. Phys. A: Math. Gen. 24 (1991), 3557-3574.
  4. Manning M.F., Rosen N., Potential functions for vibration of diatomic molecules, Phys. Rev. 44 (1933), 951-954.
  5. Wu J., Alhassid Y., The potential group approach and hypergeometric differential equations, J. Math. Phys. 31 (1990), 557-562.
    Wu J., Alhassid Y., Gürsey F., Group theory approach to scattering. IV. Solvable potentials associated with SO(2,2), Ann. Physics 196 (1989), 163-181.
  6. Levai G., Solvable potentials associated with su(1,1) algebras: a systematic study, J. Phys. A: Math. Gen. 27 (1994), 3809-3828.
  7. Cordero P., Salamó S., Algebraic solution for the Natanzon hypergeometric potentials, J. Math. Phys. 35 (1994), 3301-3307.
  8. Cordriansky S., Cordero P., Salamó S., On the generalized Morse potential, J. Phys. A: Math. Gen. 32 (1999), 6287-6293.
  9. Gangopadhyaya A., Mallow J.V., Sukhatme U.P., Translational shape invariance and inherent potential algebra, Phys. Rev. A 58 (1998), 4287-4292.
  10. 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), 111-134.
  11. Kalnins E.G., Miller W. Jr., Pogosyan G., Superintegrability on the two-dimensional hyperboloid, J. Math. Phys. 38 (1997), 5416-5433.
    Berntson B.K., Classical and quantum analogues of the Kepler problem in non-Euclidean geometries of constant curvature, B.Sc. Thesis, University of Minnesota, 2011.
  12. Gazeau J.-P., Coherent states in quantum physics, Wiley-VCH, Weinheim, 2009.
  13. Bogdanova I., Vandergheynst P., Gazeau J.-P., Continuous wavelet transformation on the hyperboloid, Appl. Comput. Harmon. Anal. 23 (2007), 286-306.
  14. Kim Y.S., Noz M.E., Theory and application of the Poincaré group, D. Reidel Publishing Co., Dordrecht, 1986.
  15. De R., Dutt R., Sukhatme U., Mapping of shape invariant potentials under point canonical transformations, J. Phys. A: Math. Gen. 25 (1992), L843-L850.
  16. Alvarez-Castillo D. E., Compean C.B., Kirchbach M., Rotational symmetry and degeneracy: a cotangent perturbed rigid rotator of unperturbed level multiplicity, Mol. Phys. 109 (2011), 1477-1483, arXiv:1105.1354.
  17. Raposo A., Weber H.-J., Alvarez-Castillo D.E., Kirchbach M., Romanovski polynomials in selected physics problems, Cent. Eur. J. Phys. 5 (2007), 253-284, arXiv:0706.3897.
  18. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.

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