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

SIGMA 9 (2013), 003, 25 pages      arXiv:1210.4515
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

From Quantum AN to E8 Trigonometric Model: Space-of-Orbits View

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

Received September 21, 2012, in final form January 11, 2013; Published online January 17, 2013

A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (ABCD)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (GFE)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC1≡(Z2)⊕T symmetry. In particular, the BC1 origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕sl(2).

Key words: (quasi)-exact-solvability; space of orbits; trigonometric models; algebraic forms; Coxeter (Weyl) invariants; hidden algebra.

pdf (529 kb)   tex (105 kb)


  1. Boreskov K.G., López Vieyra J.C., Turbiner A.V., Solvability of F4 integrable system, Internat. J. Modern Phys. A 16 (2001), 4769-4801, hep-th/0108021.
  2. Boreskov K.G., Turbiner A.V., López Vieyra J.C., Sutherland-type trigonometric models, trigonometric invariants, and multivariate polynomials, in Special Functions and Orthogonal Polynomials, Contemp. Math., Vol. 471, Amer. Math. Soc., Providence, RI, 2008, 15-31, arXiv:0805.0770.
  3. Boreskov K.G., Turbiner A.V., López Vieyra J.C., García M.A.G., Sutherland-type trigonometric models, trigonometric invariants and multivariate polynomials. III. E8 case, Internat. J. Modern Phys. A 26 (2011), 1399-1437, arXiv:1012.1902.
  4. Brink L., Turbiner A.V., Wyllard N., Hidden algebras of the (super) Calogero and Sutherland models, J. Math. Phys. 39 (1998), 1285-1315, hep-th/9705219.
  5. Chryssomalakos C., Turbiner A.V., Canonical commutation relation preserving maps, J. Phys. A: Math. Gen. 34 (2001), 10475-10485, math-ph/0104004.
  6. García M.A.G., Turbiner A.V., Hidden algebra of Hamiltonian reduction, unpublished.
  7. García M.A.G., Turbiner A.V., The quantum H3 integrable system, Internat. J. Modern Phys. A 25 (2010), 5567-5594, arXiv:1007.0737.
  8. García M.A.G., Turbiner A.V., The quantum H4 integrable system, Modern Phys. Lett. A 26 (2011), 433-447, arXiv:1011.2127.
  9. González-López A., Kamran N., Olver P.J., Lie algebras of differential operators in two complex variables, Amer. J. Math. 114 (1992), 1163-1185.
  10. Khastgir S.P., Pocklington A.J., Sasaki R., Quantum Calogero-Moser models: integrability for all root systems, J. Phys. A: Math. Gen. 33 (2000), 9033-9064, hep-th/0005277.
  11. Lie S., Gruppenregister, Vol. 5, B.G. Teubner, Leipzig, 1924.
  12. López Vieyra J.C., García M.A.G., Turbiner A.V., Sutherland-type trigonometric models, trigonometric invariants and multivariable polynomials. II. E7 case, Modern Phys. Lett. A 24 (2009), 1995-2004, arXiv:0904.0484.
  13. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  14. Oshima T., Completely integrable systems associated with classical root systems, SIGMA 3 (2007), 061, 50 pages, math-ph/0502028.
  15. Rosenbaum M., Turbiner A.V., Capella A., Solvability of the G2 integrable system, Internat. J. Modern Phys. A 13 (1998), 3885-3903, hep-th/9606092.
  16. Rühl W., Turbiner A.V., Exact solvability of the Calogero and Sutherland models, Modern Phys. Lett. A 10 (1995), 2213-2221, hep-th/9506105.
  17. Smirnov Yu., Turbiner A.V., Lie algebraic discretization of differential equations, Modern Phys. Lett. A 10 (1995), 1795-1802, Errata, Modern Phys. Lett. A 10 (1995), 3139, funct-an/9501001.
  18. Sutherland B., Exact results for a quantum many-body problem in one dimension, Phys. Rev. A 4 (1971), 2019-2021.
  19. 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.
  20. Turbiner A.V., BC2 Lame polynomials, Talks presented at 1085 Special Session of American Mathematical Society (Tucson, 2012) and Annual Meeting of Canadian Mathematical Society (Montréal, 2012).
  21. Turbiner A.V., From quantum AN (Calogero) to H4 (rational) model, SIGMA 7 (2011), 071, 20 pages, arXiv:1106.5017.
  22. Turbiner A.V., Particular integrability and (quasi)-exact-solvability, J. Phys. A: Math. Theor. 46 (2013), 025203, 9 pages, arXiv:1206.2907.
  23. Turbiner A.V., Quasi-exactly-solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), 467-474.
  24. Turbiner A.V., Two-body elliptic model in proper variables: Lie algebraic forms and their discretizations, in Calogero-Moser-Sutherland Models (Montréal, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 473-484, solv-int/9710004.

Previous article  Next article   Contents of Volume 9 (2013)