A q-difference Baxter's operator for the Ablowitz-Ladik chain

TL;DR

This paper constructs a q-difference Baxter's operator for the quantum Ablowitz-Ladik chain, connecting classical Backlund transformations with quantum integrability techniques like Bethe ansatz.

Contribution

It introduces a novel q-difference Baxter's operator for the quantum Ablowitz-Ladik model, linking classical and quantum integrability methods.

Findings

Derived the Baxter's equation as a q-difference equation.

Established the spectrality property relating classical and quantum forms.

Provided an explicit q-integral representation of the Baxter's operator.

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 Backlund 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 Backlund 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.