Noncompact SL(2,R) spin chain

TL;DR

This paper constructs and analyzes the integrable noncompact SL(2,R) spin chain, deriving key operators, spectral properties, and the solution to the spectral problem using the separation of variables method.

Contribution

It provides explicit constructions of the R-matrix, Baxter Q-operator, and transition kernel for the noncompact SL(2,R) spin chain, advancing understanding of its spectral properties.

Findings

Derived explicit expressions for energy and quasimomentum in terms of the Baxter Q-operator.

Established the analytic properties of Baxter Q-operator eigenvalues.

Solved the spectral problem using the separation of variables approach.

Abstract

We consider the integrable spin chain model - the noncompact SL(2,R) spin magnet. The spin operators are realized as the generators of the unitary principal series representation of the SL(2,R) group. In an explicit form, we construct R-matrix, the Baxter Q-operator and the transition kernel to the representation of the Separated Variables (SoV). The expressions for the energy and quasimomentum of the eigenstates in terms of the Baxter Q-operator are derived. The analytic properties of the eigenvalues of the Baxter operator as a function of the spectral parameter are established. Applying the diagrammatic approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into a product of certain operators each depending on a single separated variable.