Noncompact SL(2, ℝ) spin chain

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. © SISSA/ISAS 2004.