[Paper Review] A bispectral q-hypergeometric basis for a class of quantum integrable models

Pascal Baseilhac, Xavier Martín|arXiv (Cornell University)|Jun 23, 2015

Nonlinear Waves and Solitons81 references3 citations

TL;DR

This paper constructs a bispectral basis of multivariable $q$-orthogonal polynomials—specifically, Gasper and Rahman’s $N$-variable $q$-hypergeometric polynomials—for quantum integrable models associated with the $q$-Onsager algebra. It establishes that these polynomials form infinite-dimensional modules, realize raising/lowering operators, and yield finite-dimensional submodules under discrete spectral conditions, linking them to tridiagonal pairs and the $q$-Dolan-Grady hierarchy with generalized Nepomechie relations.

ABSTRACT

For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with $N$ variables and $N+3$ parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the $q-$Onsager algebra, whose fundamental generators are realized in terms of the multivariable $q-$difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the $N$ variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the $q-$Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case $q=1$. This framework provides a $q-$hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions ($q eq 1$).

Motivation & Objective

• To construct a bispectral basis of multivariable $q$-orthogonal polynomials for quantum integrable models governed by the $q$-Onsager algebra.
• To demonstrate that Gasper and Rahman’s $N$-variable $q$-hypergeometric polynomials generate infinite-dimensional modules for the $q$-Onsager algebra via $q$-difference operators from Iliev.
• To extend Sahi’s raising and lowering operators to the multivariable setting within the $q$-Onsager algebra framework.
• To identify finite-dimensional submodules when the $N$ variables lie on a discrete support, linking the bispectral property to tridiagonal pairs.
• To analyze eigenfunctions of the $q$-Dolan-Grady hierarchy in the polynomial basis and identify invariant subspaces under generalized Nepomechie’s relations.

Proposed method

• Utilizes Gasper and Rahman’s $N$-variable $q$-hypergeometric polynomials with $N+3$ parameters as a foundational basis for constructing modules of the $q$-Onsager algebra.
• Realizes the fundamental generators of the $q$-Onsager algebra using $q$-difference and difference operators introduced by Iliev for multivariable settings.
• Constructs multivariable raising and lowering operators extending Sahi’s single-variable construction, preserving algebraic closure under the $q$-Onsager algebra relations.
• Identifies invariant subspaces within the polynomial basis for the $q$-Dolan-Grady hierarchy under specific parameter conditions, generalizing Nepomechie’s relations.
• Analyzes the bispectral property in finite-dimensional settings by linking the structure to tridiagonal pairs, where the polynomials are eigenfunctions of two commuting tridiagonal operators.
• Extends the framework to the $q=1$ limit, connecting to known results on the $sl_n$-Onsager algebra and Tratnik polynomials, and suggests higher-rank generalizations.

Experimental results

Research questions

• RQ1 Can Gasper and Rahman’s multivariable $q$-hypergeometric polynomials serve as a bispectral basis for infinite-dimensional modules of the $q$-Onsager algebra?
• RQ2 How do the $q$-difference operators of Iliev realize the $q$-Onsager algebra generators in the multivariable setting?
• RQ3 What conditions on parameters and spectral support lead to finite-dimensional submodules with a natural tridiagonal pair interpretation?
• RQ4 How do the eigenfunctions of the $q$-Dolan-Grady hierarchy behave in this polynomial basis, and what invariant subspaces emerge?
• RQ5 What is the structure of the $q=1$ limit of this framework, and how does it relate to higher-rank generalizations of the $q$-Onsager algebra?

Key findings

• The multivariable Askey-Wilson polynomials with $N$ variables and $N+3$ parameters form an infinite-dimensional module for the $q$-Onsager algebra via Iliev’s $q$-difference operators.
• Raising and lowering operators extending Sahi’s construction are explicitly realized within the $q$-Onsager algebra framework, preserving the algebraic structure.
• For specific parameter choices and discrete variable support, finite-dimensional submodules are identified, and the bispectral property naturally corresponds to the tridiagonal pair framework.
• The eigenfunctions of the $q$-Dolan-Grady hierarchy are shown to preserve invariant subspaces under generalized Nepomechie’s relations, extending known results.
• In the $q=1$ limit, the framework connects to the $sl_n$-Onsager algebra and Tratnik’s multivariable polynomials, suggesting a path toward higher-rank generalizations.
• The analysis reveals a deep connection between $q$-hypergeometric orthogonal polynomials, quantum integrable models, and representation theory, particularly through the $q$-Onsager algebra and tridiagonal pairs.