Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

Abstract

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families. * Whereas a classical result of Bochner characterizes the Jacobi polynomials as "the most general orthogonal family satisfying a linear homogeneous second-order differential equation", the Askey-Wilson polynomials are known to constitute "the most general orthogonal family satisfying a linear homogeneous second-order q-difference equation" [GH96].

Stable equilibria for the roots of the symmetric continuous Hahn and Wilson polynomials

“…As before, the first part of the theorem hinges on the considerations in [12, Remarks 5.4, 5.5] with the main points summarized above in Subsection 3.2 (cf. also the proof of [12,Proposition 4.1] for a detailed check that the present Morse function is indeed radially unbounded).…”

“…To keep our presentation self-contained, let us briefly extract the main ingredients from the proof in [12] sustaining the claim that the coordinates of a critical point of the above Morse function are necessarily given by the roots of the symmetric continuous Hahn polynomial. To this end we first observe that at such a critical point:…”

“…Hypergeometric orthogonal polynomials beyond the Jacobi family generally do not fit within Stieltjes' electrostatic framework, for instance because these often satisfy second-order difference equations rather than differential equations. Still, for key examples of such families like the (symmetric) continuous Hahn and Wilson polynomials, it was recently observed that the roots can nevertheless be retrieved from the global minimum of a smooth Morse function, which was found by drawing from ideas developed in the context of the Bethe Ansatz method for the diagonalization of quantum integrable particle models [12]. In the present work it will be shown that the gradient flows associated with the pertinent Morse functions converge again exponentially to the roots in question.…”

“…Here, however, we will not pursue this connection and rather try to stick closer to the original electrostatic interpretation of Stieltjes, which-for the polynomials of the symmetric continuous Hahn, Wilson and Askey-Wilson families-turns out to be governed by the Morse functions found in Ref. [12]. A corresponding analysis of the gradient system for the roots of the Askey-Wilson polynomial has recently been carried out in Ref.…”

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

“…It is known that the Askey-Wilson algebra gives an algebraic framework for all the orthogonal polynomials of the Askey-scheme [Z91]. Also, it is known that the zeroes of the Askey-Wilson polynomials satisfy a class of Bethe ansatz equations [WZ95,DE19]. Define the Heun-Askey-Wilson element [BTVZ18,CVZ19]:…”