Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

Abstract

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pmβ' p} (β' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium positions of the corresponding Calogero and Sutherland systems. The orthogonality of the original polynomials is inherited by the deformed ones which satisfy three-term recurrence and certain functional equations. The latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials.

45 pages. A few typos in section 6 are corrected

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Cited by in corpus (13)

• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Discrete Quantum Mechanics
• Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials
• On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System
• New Quasi Exactly Solvable Difference Equation
• New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential
• Multi-Particle Quasi Exactly Solvable Difference Equations
• Perturbations around the zeros of classical orthogonal polynomials
• Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials
• Deformed multi-variable Fokker-Planck equations
• Stable equilibria for the roots of the symmetric continuous Hahn and Wilson polynomials