Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

TL;DR

This paper develops Bethe ansatz solutions for a class of quasi exactly solvable difference equations, extending known polynomial solutions and providing explicit eigenfunctions and eigenvalues.

Contribution

It introduces a Bethe ansatz framework for new quasi exactly solvable difference equations related to classical orthogonal polynomials.

Findings

Eigenfunctions are polynomials with roots satisfying Bethe ansatz equations

Eigenvalues are expressed in terms of roots of Bethe ansatz equations

Provides explicit solutions for deformations of classical orthogonal polynomial difference equations

Abstract

Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the well-known exactly solvable difference equations of the Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. Up to an overall factor of the so-called pseudo ground state wavefunction, the eigenfunctions within the exactly solvable subspace are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots.