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arXiv:1803.06196 · Reviews in Mathematical Physics

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

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

Published Jun 5, 2018• 14 citations

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

DOI: 10.1142/s0129055x18400032

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We construct the Baxter operator and the corresponding Baxter equation for a quantum version of the Ablowitz–Ladik model. The result is achieved in two different ways: by using the well-known Bethe ansatz technique and by looking at the quantum analogue of the classical Bäcklund transformations. General results about integrable models governed by the same r -matrix algebra will be given. Baxter's equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Bäcklund transformations gives a trace formula representing the classical analogue of Baxter's equation. A q-integral representation of the Baxter operator is discussed.

Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's

q

we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex

R

-matrix. Both independent Baxter's

Q

-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of

Q

-s is also calculated.