Baxter Operator and Baxter Equation for q-Toda and Toda2 Chains

Abstract

We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 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 [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev