Separation of variables for the quantum SL(2,R) spin chain

We construct a representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain following the Sklyanin's approach and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same 'pyramid diagram' as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state. (author)