Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models

TL;DR

This paper extends the analysis of integrable quantum models with cyclic representations using Sklyanin's Separation of Variables, demonstrating the completeness of the Bethe Ansatz and characterizing the Baxter Q-operator.

Contribution

It generalizes the spectrum construction for cyclic representations and proves the completeness of the Bethe Ansatz in this context.

Findings

Transfer matrix spectrum is fully characterized by polynomial solutions of Baxter equations.

The method confirms the completeness of the transfer matrix eigenstates.

Existence and characterization of the Baxter Q-operator are established.

Abstract

In [1] an integrable quantum model was introduced and a class of its cyclic representations was proven to define lattice regularizations of the Sine-Gordon model. Here, we analyze general cyclic representations of this integrable quantum model by extending the spectrum construction introduced in [2] in the framework of the Separation of Variables (SOV) of Sklyanin. We show that as in [1] also for general representations, the transfer matrix spectrum (eigenvalues and eigenstates) is completely characterized in terms of polynomial solutions of an associated functional Baxter equation. Moreover, we prove that the method here developed has two fundamental built-in features: i) the completeness of the set of the transfer matrix eigenstates constructed from the solutions of the associated Bethe ansatz equations, ii) the existence and complete characterization of the Baxter Q-operator.