A q-difference Baxter operator for the Ablowitz–Ladik chain

Abstract

We construct the Baxter's operator and the corresponding Baxter's equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Bäcklund transformations. For comparison we find the same result by using the well-known Bethe ansatz technique. General results about integrable models governed by the same r-matrix algebra will be given. The 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 the Baxter's equation. An explicit q-integral representation of the Baxter's operator is discussed.

From quantum Bäcklund transforms to topological quantum field theory

“…Remark 4.6. After the construction of Q + in [15, section 3] an alternative expression for a Q-operator has been put forward [35] in the Fock representation. In loc.…”

q-bosons, Toda lattice, Pieri rules and Baxterq-operator

“…The results reported in this section have been obtained in collaboration with Olivier Babelon and Simon Ruisjenaars [24]. The q-Toda Q-matrix (116) was obtained in a previous work with Michel Gaudin [27] and related Q-matrices appear in recent works [16,28,29].…”

“…Then, similarly as in ( 26), (28), the explicit expression of the transfer matrices R z , ( ) G  acting on the semi infinite chains is:…”

On Existence of Solutions for Nonlinear Q-Difference Equations With Nonlocal Q-Integral Boundary Conditions

“…The systematic research on q-difference equations owes to Jackson [27], Carmichael [18], Mason [33] and Adams [2] in the first quarter of 20th century. For some real applications of q-calculus, we refer the reader to the models [32,37] and the references cited therein. An important characteristic of q-difference equations is that they are always completely controllable and hence appear in the q-optimal control problems [15].…”