The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

TL;DR

This paper explores the spectral properties of the q-Onsager algebra, deriving eigenfunctions via a second-order q-difference equation, and extends analysis beyond the Bethe ansatz to construct all eigenstates of the XXZ spin chain.

Contribution

It introduces a novel approach to analyze the q-Onsager algebra, enabling the construction of all eigenstates beyond the Bethe ansatz framework for the XXZ spin chain.

Findings

Eigenfunctions satisfy a second-order q-difference equation.

Bethe equations emerge in the algebraic sector with polynomial eigenfunctions.

All eigenstates of the XXZ spin chain are explicitly constructed.

Abstract

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.