2004/10/22 by S. Odake, Satoru Odake, Ryu Sasaki +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Askey–Wilson polynomials #Classical orthogonal polynomials #Discrete orthogonal polynomials #Eigenfunction #Eigenvalues and eigenvectors #Gegenbauer polynomials #Hahn polynomials #Hermite polynomials #Invariant (physics) #Laguerre polynomials #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Trigonometry #Wilson polynomials #hep-th #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1063/1.1927080
published as J.Math.Phys. 46 (2005) 063513 · 14 pages, 1 figure. The outline of derivation of the result in section 2 is added
arxiv created 2004/10/22 · openalex publication_date 2005/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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, 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.