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

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