Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems

Abstract

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". The equilibrium positions of Calogero and Sutherland systems for the classical root systems (A_r, B_r, C_r and D_r) correspond to the zeros of Hermite, Laguerre, Jacobi and Chebyshev polynomials. Here we define and derive the corresponding polynomials for the exceptional (E_6, E_7, E_8, F_4 and G_2) and non-crystallographic (I_2(m), H_3 and H_4) root systems. They do not have orthogonality but share many other properties with the above mentioned classical polynomials.

41 pages. A Mathematica file "poly.m" is attached

References in corpus (2)

• Quantum vs Classical Integrability in Calogero-Moser Systems
• Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models

Cited by in corpus (12)

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• Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach
• The spin Sutherland model of D_N type and its associated spin chain
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Shape invariance in prepotential approach to exactly solvable models
• Affine Toda-Sutherland Systems
• Classical vs Quantum Mechanics: role of elementary excitations
• The open Haldane-Shastry chain: thermodynamics and criticality
• Polynomials Associated with Equilibria of Affine Toda-Sutherland Systems