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

Source: [Baxter operator and Baxter equation for $q$-Toda and Toda$\_2$ chains](<https://www.sota2.com/research/paper/68fa38221e209131ce8b7d7c>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T22:47:28.109Z

Authors: O. Babelon, K.K. Kozlowski, V. Pasquier

Published: 2018-03-16T12:44:31.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1803.06196>)

[PDF](<https://arxiv.org/pdf/1803.06196.pdf>)