Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials

Abstract

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of "discrete" quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are the Askey-Wilson polynomials. This is an extension of our previous study [1,2], which established the "discrete analogue" of the well-known fact; The equilibrium positions of the Calogero systems are described by the Hermite and Laguerre polynomials, whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the Hermite and Laguerre polynomials.

14 pages, 1 figure. The outline of derivation of the result in section 2 is added

References in corpus (8)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Generalized coherent and intelligent states for exact solvable quantum systems
• Multivariable continuous Hahn and Wilson polynomials related to integrable difference systems
• On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System
• Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials
• Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems
• Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems
• Bethe Ansatz Equations of XXZ Model and q-Sturm-Liouville Problems

Cited by in corpus (23)

• Orthogonal Polynomials from Hermitian Matrices
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Discrete Quantum Mechanics
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Orthogonal Polynomials from Hermitian Matrices II
• q-oscillator from the q-Hermite Polynomial
• Quasi Exactly Solvable Difference Equations
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
• Properties of the zeros of the polynomials belonging to the Askey scheme
• New Quasi Exactly Solvable Difference Equation
• Multi-Particle Quasi Exactly Solvable Difference Equations
• Shape invariance through Crum transformation
• Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations
• Perturbations around the zeros of classical orthogonal polynomials
• Finite-dimensional representations of difference operators, and the identification of remarkable matrices
• Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials
• Deformed multi-variable Fokker-Planck equations
• Interpolation of SUSY quantum mechanics
• Stable equilibria for the roots of the symmetric continuous Hahn and Wilson polynomials