Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain

We construct the Baxter Bbb Q-operator and the representation of the Separated Variables (SoV) for the homogeneous open SL(2,Bbb R) spin chain. Applying the diagrammatical 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 Bbb Q-operator. We show that the transition kernel to the SoV representation is factorized into the product of certain operators each depending on a single separated variable. As a consequence, it has a universal pyramid-like form that has been already observed for various quantum integrable models such as periodic Toda chain, closed SL(2,Bbb R) and SL(2,Bbb C) spin chains.