Baxter operator and Baxter equation for $q$-Toda and Toda$_2$ chains

TL;DR

This paper constructs the Baxter operator for $q$-Toda and Toda$_2$ chains, establishing their integrability, modular invariance, and deriving the Baxter equation, which paves the way for spectrum quantization.

Contribution

It introduces a new construction of the Baxter operator for these chains, linking it to Bäcklund transformations and ensuring modular invariance.

Findings

Constructed the Baxter operator for $q$-Toda and Toda$_2$ chains.

Proved the Baxter operator's modular invariance.

Derived the Baxter equation for eigenvalues.

Abstract

We construct the Baxter operator $\boldsymbol{ \texttt{Q} }(\lambda)$ for the $q$-Toda chain and the Toda$_2$ chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and B\"acklund transformations that were unravelled in {\cite{GaPa92}}. We construct a number of quantum intertwiners ensuring the commutativity of $\boldsymbol{ \texttt{Q} }(\lambda)$ with the transfer matrix of the models and the one of $\boldsymbol{ \texttt{Q} }$'s between each other. Most importantly, $\boldsymbol{ \texttt{Q} }(\lambda)$ is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues $q(\lambda)$ of $\boldsymbol{ \texttt{Q} }(\lambda)$ and show that these are entire functions of $\lambda$. This last property will ultimately lead to the quantisation of the spectrum for the considered Toda chains, in a subsequent publication.