Noncompact SL(2,R) spin chain

We consider the integrable spin chain model - the noncompact SL(2,R) spinmagnet. The spin operators are realized as the generators of the unitaryprincipal series representation of the SL(2,R) group. In an explicit form, weconstruct R-matrix, the Baxter Q-operator and the transition kernel to therepresentation of the Separated Variables (SoV). The expressions for the energyand quasimomentum of the eigenstates in terms of the Baxter Q-operator arederived. The analytic properties of the eigenvalues of the Baxter operator as afunction of the spectral parameter are established. Applying the diagrammaticapproach, we calculate Sklyanin's integration measure in the separatedvariables and obtain the solution to the spectral problem for the model interms of the eigenvalues of the Q-operator. We show that the transition kernelto the SoV representation is factorized into a product of certain operatorseach depending on a single separated variable.Comment: 29 pages, 12 figure