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Mathematical Physics

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

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.
Comments: 31 pages
Subjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
Cite as: arXiv:1803.06196 [math-ph]
  (or arXiv:1803.06196v1 [math-ph] for this version)

Submission history

From: Karol Kozlowski Kajetan [view email]
[v1] Fri, 16 Mar 2018 12:44:31 GMT (30kb)