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

SIGMA 5 (2009), 039, 23 pages      arXiv:0904.0170      https://doi.org/10.3842/SIGMA.2009.039
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Intertwining Symmetry Algebras of Quantum Superintegrable Systems

Juan A. Calzada a, Javier Negro b and Mariano A. del Olmo b
a) Departamento de Matemática Aplicada, Universidad de Valladolid, E-47011, Valladolid, Spain
b) Departamento de Física Teórica, Universidad de Valladolid, E-47011, Valladolid, Spain

Received November 14, 2008, in final form March 18, 2009; Published online April 01, 2009

Abstract
We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n),so(2n)) or (su(p,q),so(2p,2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness.

Key words: superintegrable systems; intertwining operators; dynamical algebras.

pdf (905 kb)   ps (1076 kb)   tex (1465 kb)

References

  1. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666-5676.
    Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483-486.
    Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.
  2. Bonatsos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
  3. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  4. Grosche C., Pogosyan G.S., Sisakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials. II. The two- and three-dimensional sphere, Fortschr. Phys. 43 (1995), 523-563, hep-th/9402121.
    Grosche C., Pogosyan G.S., Sisakian A.N., Path integral approach to superintegrable potentials. III. Two-dimensional hyperboloid, Phys. Particles Nuclei 27 (1996), 244-272.
    Grosche C., Pogosyan G.S., Sisakian A.N., Path integral discussion for superintegrable potentials. IV. Three dimensional pseudosphere, Phys. Particles Nuclei 28 (1997), 486-519.
  5. Ballesteros A., Herranz F.J., Santander M., Sanz-Gil T., Maximal superintegrability on N-dimensional curved spaces, J. Phys. A: Math. Gen. 36 (2003), L93-L99, math-ph/0211012.
  6. Cariñena J.F., Rañada M.F., Santander M., Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys. 46 (2005), 052702, 25 pages, math-ph/0504016.
    Cariñena J.F., Rañada M.F., Santander M., Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton, J. Phys. A: Math. Theor. 40 (2007), 13645-13666.
  7. Lakshmanan M., Eswaran K., Quantum dynamics of a solvable nonlinear chiral model, J. Phys. A: Math. Gen. 8 (1975), 1658-1669.
  8. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.
  9. Marsden J.E., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  10. del Olmo M.A., Rodríguez M.A., Winternitz P., Integrable systems based on SU(p,q) homogeneous manifolds, J. Math. Phys. 34 (1993), 5118-5139.
  11. del Olmo M.A., Rodríguez M. A., Winternitz P., The conformal group SU(2,2) and integrable systems on a Lorentzian hyperboloid, Fortschr. Phys. 44 (1996), 199-233, hep-th/9407080.
  12. Calzada J.A., del Olmo M.A., Rodríguez M.A., Classical superintegrable SO(p,q) Hamiltonian systems, J. Geom. Phys. 23 (1997), 14-30.
  13. Calzada J.A., del Olmo M.A., Rodríguez M.A., Pseudo-orthogonal groups and integrable dynamical systems in two dimensions, J. Math. Phys. 40 (1999), 188-209, solv-int/9810010.
  14. Calzada J.A., Negro J., del Olmo M.A., Rodríguez M.A., Contraction of superintegrable Hamiltonian systems, J. Math. Phys. 41 (1999), 317-336.
  15. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  16. Alhassid Y., Gürsey F., Iachello F., Group theory approach to scattering, Ann. Physics 148 (1983), 346-380.
  17. Kuru S., Tegmen A., Vercin A., Intertwined isospectral potentials in an arbitrary dimension, J. Math. Phys. 42 (2001), 3344-3360, quant-ph/0111034.
  18. Demircioglu B., Kuru S., Önder M., Vercin A., Two families of superintegrable and isospectral potentials in two dimensions, J. Math. Phys. 43 (2002), 2133-2150, quant-ph/0201099.
  19. Fernández D.J., Negro J., del Olmo M.A., Group approach to the factorization of the radial oscillator equation, Ann. Physics 252 (1996), 386-412.
  20. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
  21. Calzada J.A., Negro J., del Olmo M.A., Superintegrable quantum u(3) systems and higher rank factorizations, J. Math. Phys. 47 (2006), 043511, 17 pages, math-ph/0601067.
  22. Calzada J.A., Kuru S., Negro J., del Olmo M.A., Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid, J. Phys. A: Math. Theor. 41 (2008), 255201, 11 pages, arXiv:0803.2117.
  23. Kobayashi S., Nomizu K., Foundations of differential geometry, Interscience Publishers, New York - London, 1963.
  24. del Olmo M.A., Rodríguez M.A., Winternitz P., Zassenhaus H., Maximal abelian subalgebras of pseudounitary Lie algebras, Linear Algebra Appl. 135 (1990), 79-151.
  25. Zhedanov A.S., The "Higgs algebra" as a "quantum" deformation of SU(2), Modern Phys. Lett. A 7 (1992), 507-512.
  26. Bambah B.A., Sunil Kumar V., Mukku C., Polynomial algebras: their representations, coherent states and applications to quantum mechanics, J. Theor. Phys. Group Theory Nonlinear Opt. 11 (2007), 265-284.
  27. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions, J. Math. Phys. 37 (1996), 6439-6467.
  28. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid, J. Math. Phys. 38 (1997), 5416-5433.
  29. Grabowski J., Landi G., Marmo G., Vilasi G., Generalized reduction procedure: symplectic and Poisson formalism, Fortschr. Phys. 42 (1994), 393-427, hep-th/9307018.

Previous article   Next article   Contents of Volume 5 (2009)