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

Abstract

We construct the Baxter operator $\text{Q}(\lambda )$ 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 Bäcklund transformations that were unravelled in {\cite{GaPa92}}. We construct a number of quantum intertwiners ensuring the commutativity of $\text{Q}(\lambda )$ with the transfer matrix of the models and the one of $\text{Q}$'s between each other. Most importantly, $\text{Q}(\lambda )$ is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues $q(\lambda )$ of $\text{Q}(\lambda )$ 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.

arXiv comment31 pages

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Cited by in corpus (6)

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• On the spectrum of the local ${\mathbb{P}}^2$ mirror curve
• Solution of Baxter equation for the $q$-Toda and Toda${}_2$ chains by NLIE
• On the Q operator and the spectrum of the XXZ model at root of unity