Theoretical and Mathematical Physics

, Volume 191, Issue 1, pp 480–490 | Cite as

Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians



We hypothesize the form of a transformation reducing the elliptic A N Calogero–Moser operator to a differential operator with polynomial coefficients. We verify this hypothesis for N ≤ 3 and, moreover, give the corresponding polynomial operators explicitly.


elliptic Calogero–Moser Hamiltonian universal enveloping algebra 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1996).CrossRefMATHGoogle Scholar
  2. 2.
    V. V. Sokolov and A. V. Turbiner, “Quasi-exact-solvability of the A2/G2 elliptic model: Algebraic form, sl(3)/g(2) hidden algebra, and polynomial eigenfunctions,” J. Phys. A: Math. Theor., 48, 155201 (2015); arXiv:1409.7439v2 [math-ph] (2014).ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    W. Rühl and A. V. Turbiner, “Exact solvability of the Calogero and Sutherland models,” Modern Phys. Lett. A, 10, 2213–2222 (1995).ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    G. Post and N. van den Hijligenberg, “gl(?) and differential operators preserving polynomials,” Acta Appl. Math., 44, 257–268 (1996).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge (1997).CrossRefMATHGoogle Scholar
  6. 6.
    H. Awata, Y. Matsuo, S. Odake, and J. Shiraishi, “Collective field theory, Calogero–Sutherland model, and generalized matrix models,” Phys. Lett. B, 347, 49–55 (1995).ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. N. Sergeev and A. P. Veselov, “Dunkl operators at infinity and Calogero–Moser systems,” Internat. Math. Res. Notices, 2015, 10959–10986; arXiv:1311.0853v2 [math-ph] (2013).CrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.National Research University “Higher School of Economics,”MoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia

Personalised recommendations