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

SIGMA 11 (2015), 096, 23 pages      arXiv:1504.06228      https://doi.org/10.3842/SIGMA.2015.096
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Harmonic Oscillator on the ${\rm SO}(2,2)$ Hyperboloid

Davit R. Petrosyan a and George S. Pogosyan bc
a) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia
b) Departamento de Matematicas, CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico
c) International Center for Advanced Studies, Yerevan State University, A. Manoogian 1, Yerevan, 0025, Armenia

Received April 24, 2015, in final form November 20, 2015; Published online November 25, 2015

Abstract
In the present work the classical problem of harmonic oscillator in the hyperbolic space $H_2^2$: $z_0^2+z_1^2-z_2^2-z_3^2=R^2$ has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on $H_2^2$, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones.

Key words: superintegrable systems; harmonic oscillator; hyperbolic space; Hamilton-Jacobi equation.

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References

  1. Ambrozio L.C., On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass, Comm. Math. Phys. 337 (2015), 767-783, arXiv:1402.4317.
  2. Bogush A.A., Kurochkin Yu.A., Otchik V.S., The quantum-mechanical Kepler problem in three-dimensional Lobachevsky space, Dokl. Akad. Nauk BSSR 24 (1980), 19-22.
  3. Bracken P.F., Hamiltonians for the quantum Hall effect on spaces with non-constant metrics, Internat. J. Theoret. Phys. 46 (2007), 119-132, math-ph/0607051.
  4. Calzada J.A., del Olmo M.A., Rodríguez M.A., Classical superintegrable ${\rm SO}(p,q)$ Hamiltonian systems, J. Geom. Phys. 23 (1997), 14-30.
  5. Cariñena J.F., Perelomov A.M., Rañada M.F., sochronous classical systems and quantum systems with equally space spectra, J. Phys. Conf. Ser. 87 (2007), 012007, 14 pages.
  6. Cariñena J.F., Rañada M.F., Santander M., The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature, J. Math. Phys. 49 (2008), 032703, 27 pages, arXiv:0709.2572.
  7. Chernikov N.A., The Kepler problem in the Lobachevsky space and its solution, Acta Phys. Polon. B 23 (1992), 115-122.
  8. del Olmo M.A., Rodríguez M.A., Winternitz P., The conformal group ${\rm SU}(2,2)$ and integrable systems on a Lorentzian hyperboloid, Fortschr. Phys. 44 (1996), 199-233, hep-th/9407080.
  9. Demkov Yu.N., Symmetry group of the isotropic oscillator, Soviet Phys. JETP 9 (1959), 63-66.
  10. Dombrowski P., Zitterbarth J., On the planetary motion in the $3$-dim. standard spaces ${\bf M}^3_\kappa$ of constant curvature $\kappa\in{\bf R}$, Demonstratio Math. 24 (1991), 375-458.
  11. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666-5676.
  12. Fradkin D.M., Existence of the dynamical symmetries ${\rm O}_4$ and ${\rm SU}_3$ for all classical central potential problems, Progr. Theoret. Phys. 37 (1967), 798-812.
  13. Gazeau J.-P., Piechocki W., Coherent state quantization of a particle in de Sitter space, J. Phys. A: Math. Gen. 37 (2004), 6977-6986, hep-th/0308019.
  14. Gibbons G.W., Anti-de-Sitter spacetime and its uses, in Mathematical and Quantum Aspects of Relativity and Cosmology (Pythagoreon, 1998), Lecture Notes in Phys., Vol. 537, Springer, Berlin, 2000, 102-142, arXiv:1110.1206.
  15. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, Academic Press, New York, 1980.
  16. Grosche C., On the path integral in imaginary Lobachevsky space, J. Phys. A: Math. Gen. 27 (1994), 3475-3489, hep-th/9310162.
  17. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials. I. Two- and three-dimensional Euclidean space, Fortschr. Phys. 43 (1995), 453-521, hep-th/9402121.
  18. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials. II. The two- and three-dimensional sphere, Fortschr. Phys. 43 (1995), 523-563.
  19. Grosche C., Pogosyan G.S., Sissakian A.N., Interbasis expansion for the Kaluza-Klein monopole system, in Symmetry Methods in Physics, Vol. 1 (Dubna, 1995), Joint Inst. Nuclear Res., Dubna, 1996, 245-254.
  20. Grosche C., Pogosyan G.S., Sissakian A.N., Path-integral approach for superintegrable potentials on the three-dimensional hyperboloid, Phys. Part. Nuclei 28 (1997), 486-519.
  21. Hakobyan Ye.M., Pogosyan G.S., Sissakian A.N., Vinitsky S.I., Isotropic oscillator in the space of constant positive curvature. Interbasis expansions, Phys. Atomic Nuclei 62 (1999), 623-637, quant-ph/9710045.
  22. Herranz F.J., Ballesteros Á., Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006), 010, 22 pages, math-ph/0512084.
  23. Herranz F.J., Ballesteros Á., Santander M., Sanz-Gil T., Maximally superintegrable Smorodinsky-Winternitz systems on the $N$-dimensional sphere and hyperbolic spaces, in Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, Vol. 37, Editors P. Tempesta, P. Winternitz, J. Harnad, W. Miller Jr., G. Pogosyan, M.A. Rodrigues, Amer. Math. Soc., Providence, RI, 2004, 75-89, math-ph/0501035.
  24. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.
  25. Infeld L., Schild A., A note on the Kepler problem in a space of constant negative curvature, Phys. Rev. 67 (1945), 121-122.
  26. Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics, Phys. Rev. 57 (1940), 641-645.
  27. Kalnins E.G., Kress, Miller Jr. W., Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 092001, 8 pages, arXiv:0912.3158.
  28. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
  29. Kalnins E.G., Miller Jr. W., The wave equation, ${\rm O}(2, 2)$, and separation of variables on hyperboloids, Proc. Roy. Soc. Edinburgh Sect. A 79 (1978), 227-256.
  30. Kalnins E.G., Miller Jr. W., Hakobyan Ye.M., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid. II, J. Math. Phys. 40 (1999), 2291-2306, quant-ph/9907037.
  31. Kalnins E.G., Miller Jr. W., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid, J. Math. Phys. 38 (1997), 5416-5433.
  32. Kalnins E.G., Miller Jr. W., Pogosyan G.S., The Coulomb-oscillator relation on $n$-dimensional spheres and hyperboloids, Phys. Atomic Nuclei 65 (2002), 1086-1094, math-ph/0210002.
  33. Kotecha V., Ward R.S., Integrable Yang-Mills-Higgs equations in three-dimensional de Sitter space-time, J. Math. Phys. 42 (2001), 1018-1025, nlin.SI/0012004.
  34. Kozlov V.V., Harin A.O., Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom. 54 (1992), 393-399.
  35. Landau L.D., Lifshitz E.M., Course of theoretical physics. Vol. 1. Mechanics, 3rd ed., Pergamon Press, Oxford - New York - Toronto, 1976.
  36. Leemon H.I., Dynamical symmetries in a spherical geometry. II, J. Phys. A: Math. Gen. 12 (1979), 489-501.
  37. Liebmann H., Nichteuklidische Geometrie. 3. Auflage, W. de Gruyter & Co, Berlin, 1923.
  38. Lobachevsky N.I., Complete collected works. Vol. 2, GITTL, Moscow, 1949.
  39. Makarov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries. I. The integrals of motion, Nuovo Cimento A 52 (1967), 1061-1084.
  40. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  41. Olevskii M.N., Triorthogonal systems in spaces of constant curvature in which the equation $\Delta_2u+\lambda u=0$ allows a complete separation of variables, Mat. Sb. 27 (1950), 379-426.
  42. Parker L.E., Toms D.J., Quantum field theory in curved spacetime. Quantized fields and gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2009.
  43. Petrosyan D.R., Pogosyan G.S., The Kepler-Coulomb problem on ${\rm SO}(2,2)$ hyperboloid, Phys. Atomic Nuclei 75 (2012), 1272-1278.
  44. Petrosyan D.R., Pogosyan G.S., Classical Kepler-Coulomb problem on ${\rm SO}(2,2)$ hyperboloid, Phys. Atomic Nuclei 76 (2013), 1273-1283.
  45. Petrosyan D.R., Pogosyan G.S., Oscillator problem on ${\rm SO}(2,2)$ hyperboloid, Nonlinear Phenom. Complex Syst. 17 (2014), 405-408.
  46. Schrödinger E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A. 46 (1940), 9-16.
  47. Sławianowski J.J., Bertrand systems on spaces of constant sectional curvature. The action-angle analysis, Rep. Math. Phys. 46 (2000), 429-460.
  48. 't Hooft G., Nonperturbative $2$ particle scattering amplitudes in $2+1$-dimensional quantum gravity, Comm. Math. Phys. 117 (1988), 685-700.
  49. Varlamov V.V., CPT groups of spinor fields in de Sitter and anti-de Sitter spaces, Adv. Appl. Clifford Algebr. 25 (2015), 487-516, arXiv:1401.7723.
  50. Winternitz P., Smorodinskiǐ Ya.A., Uhlíř M., Friš I., Symmetry groups in classical and quantum mechanics, Soviet J. Nuclear Phys. 4 (1967), 444-450.
  51. Zhou Z., Solutions of the Yang-Mills-Higgs equations in $(2+1)$-dimensional anti-de Sitter space-time, J. Math. Phys. 42 (2001), 1085-1099, nlin/0011020.

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