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


SIGMA 7 (2011), 071, 20 pages      arXiv:1106.5017      https://doi.org/10.3842/SIGMA.2011.071
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

From Quantum AN (Calogero) to H4 (Rational) Model

Alexander V. Turbiner
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico

Received February 28, 2011, in final form July 12, 2011; Published online July 18, 2011

Abstract
A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).

Key words: (quasi)-exact-solvability; rational models; algebraic forms; Coxeter (Weyl) invariants, hidden algebra.

pdf (463 kb)   tex (37 kb)

References

  1. Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.
  2. Turbiner A.V., Quasi-exactly-solvable problems and the sl(2) group, Comm. Math. Phys. 118 (1988), 467-474.
  3. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
    Calogero F., Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  4. Rühl W., Turbiner A., Exact solvability of the Calogero and Sutherland models, Modern Phys. Lett. A 10 (1995), 2213-2221, hep-th/9506105.
  5. Minzoni A., Rosenbaum M., Turbiner A., Quasi-exactly-solvable many-body problems, Modern Phys. Lett. A 11 (1996), 1977-1984, hep-th/9606092.
  6. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  7. Oshima T., Completely integrable systems associated with classical root systems, SIGMA 3 (2007), 061, 50 pages, math-ph/0502028.
  8. Brink L., Turbiner A., Wyllard N., Hidden algebras of the (super) Calogero and Sutherland models, J. Math. Phys. 39 (1998), 1285-1315, hep-th/9705219.
  9. Wolfes J., On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974), 1420-1424.
  10. Lie S., Gruppenregister, Vol. 5, B.G. Teubner, Leipzig, 1924, 767-773.
  11. González-López A., Kamran N., Olver P.J., Quasi-exactly-solvable Lie algebras of the first order differential operators in two complex variables, J. Phys. A: Math. Gen. 24 (1991), 3995-4008.
    González-López A., Kamran N., Olver P.J., Lie algebras of differential operators in two complex variables, Amer. J. Math. 114 (1992), 1163-1185.
  12. Boreskov K.G.,Turbiner A.V., Lopez Vieyra J.C., Solvability of the Hamiltonians related to exceptional root spaces: rational case, Comm. Math. Phys. 260 (2005), 17-44, hep-th/0407204.
  13. Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages, arXiv:0904.0738.
  14. García M.A.G., Turbiner A.V., The quantum H3 integrable system, Internat. J. Modern Phys. A 25 (2010), 5567-5594, arXiv:1007.0737.
  15. García M.A.G., Turbiner A.V., The quantum H4 integrable system, Modern Phys. Lett. A 26 (2011), 433-447, arXiv:1011.2127.
  16. García M.A.G. Los Sistemas Integrables H3 y H4, PhD Thesis, UNAM, 2011 (in Spanish).

Previous article   Next article   Contents of Volume 7 (2011)