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


SIGMA 3 (2007), 037, 17 pages      math-ph/0703012      https://doi.org/10.3842/SIGMA.2007.037
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

An Explicit Formula for Symmetric Polynomials Related to the Eigenfunctions of Calogero-Sutherland Models

Martin Hallnäs
Department of Theoretical Physics, Albanova University Center, SE-106 91 Stockholm, Sweden

Received November 01, 2006, in final form February 05, 2007; Published online March 01, 2007

Abstract
We review a recent construction of an explicit analytic series representation for symmetric polynomials which up to a groundstate factor are eigenfunctions of Calogero-Sutherland type models. We also indicate a generalisation of this result to polynomials which give the eigenfunctions of so-called 'deformed' Calogero-Sutherland type models.

Key words: quantum integrable systems; orthogonal polynomials; symmetric functions.

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References

  1. Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
  2. Calogero F., Groundstate of a one-dimensional N-body system, J. Math. Phys. 10 (1969), 2197-2200.
  3. Calogero F., Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  4. Chalykh O., Feigin M., Veselov A., New integrable generalizations of Calogero-Moser quantum problems, J. Math. Phys. 39 (1998), 695-703.
  5. Desrosiers P., Lapointe L., Mathieu P., Explicit formulas for the generalized Hermite polynomials in superspace, J. Phys. A: Math. Gen. 37 (2004), 1251-1268, hep-th/0309067.
  6. Gómez-Ullate D., González-López A., Rodríguez M.A., New algebraic quantum many-body problems, J. Phys. A: Math. Gen. 33 (2000), 7305-7335, nlin.SI/0003005.
  7. Hallnäs M., Langmann E., Explicit formulae for the eigenfunctions of the N-body Calogero model, J. Phys. A: Math. Gen. 39 (2006), 3511-3533, math-ph/0511040.
  8. Hallnäs M., Langmann E., Quantum Calogero-Sutherland type models and generalised classical polynomials, in preparation.
  9. Knop F., Sahi S., A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9-22, q-alg/9610016.
  10. Kuznetsov V.B., Mangazeev V.V., Sklyanin E.K., Q-operator and factorised separation chain for Jack polynomials, Indag. Math. (N.S.) 14 (2003), 451-482, math.CA/0306242.
  11. Langmann E., Algorithms to solve the (quantum) Sutherland model, J. Math. Phys. 42 (2001), 4148-4157, math-ph/0104039.
  12. Langmann E., A method to derive explicit formulas for an elliptic generalization of the Jack polynomials, in Proceedings of the Conference "Jack, Hall-Littlewood and Macdonald Polynomials" (September 23-26, 2003, Edinburgh), Editors V.B. Kuznetsov and S. Sahi, Contemp. Math. 417 (2006), 257-270, math-ph/0511015.
  13. Lassalle M., Polynômes de Jacobi généralisés, C. R. Math. Acad. Sci. Paris 312 (1991), 425-428.
  14. Lassalle M., Polynômes de Laguerre généralisés, C. R. Math. Acad. Sci. Paris 312 (1991), 725-728.
  15. Lassalle M., Polynômes de Hermite généralisés, C. R. Math. Acad. Sci. Paris 313 (1991), 579-582.
  16. Lassalle M., Schlosser M., Inversion of the Pieri formula for MacDonald polynomials, Adv. Math. 202 (2006), 289-325, math.CO/0402127.
  17. MacDonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, 1995.
  18. MacDonald I.G., Hypergeometric functions, unpublished manuscript.
  19. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  20. Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, 1975.
  21. Sergeev A.N., Veselov A.P., Deformed quantum Calogero-Moser problems and Lie superalgebras, Comm. Math. Phys. 245 (2004), 249-278, math-ph/0303025.
  22. Sergeev A.N., Veselov A.P., Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials, Adv. Math. 192 (2005), 341-375, math-ph/0307036.
  23. Sutherland B., Quantum many-body problem in one dimension: ground state, J. Math. Phys. 12 (1971), 246-250.
  24. Sutherland B., Exact results for a quantum many-body problem in one dimension: II, Phys. Rev. A 5 (1972), 1372-1376.
  25. Szegö G., Orthogonal polynomials, American Mathematical Society, 1939.
  26. van Diejen J.F., Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), 467-497, q-alg/9609032.
  27. Lapointe L., Morse J., van Diejen J.F., Determinantal construction of orthogonal polynomials associated with root systems, Compos. Math. 140 (2004), 255-273, math.CO/0303263.


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