Algebraization of difference eigenvalue equations related to $U_q(sl_2)$

TL;DR

This paper introduces a class of second order difference operators linked to the quantum algebra $U_q(sl_2)$, revealing algebraic spectrum parts with polynomial eigenfunctions and connections to Bethe Ansatz equations, with applications in physics.

Contribution

It presents a novel algebraic framework for difference eigenvalue equations using $U_q(sl_2)$, including polynomial solutions and spectral analysis.

Findings

Eigenfunctions are discrete polynomials.

Roots of polynomials satisfy Bethe Ansatz equations.

Applications to Bloch electrons in magnetic fields.

Abstract

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators can be constructed by means of $U_q(sl_2)$, the quantum deformation of the $sl_2$ algebra. The roots of polinomials determine the spectrum and obey the Bethe Ansatz equations. A particular case of difference equations for $q$-hypergeometric and Askey-Wilson polinomials is discussed. Applications to the problem of Bloch electrons in magnetic field are outlined. {abstract}