  In [1], the spectrum (eigenvalues and eigenstates) of a lattice
regularizations of the Sine-Gordon model has been completely characterized in
terms of polynomial solutions with certain properties of the Baxter equation.
This characterization for cyclic representations has been derived by the use of
the Separation of Variables (SOV) method of Sklyanin and by the direct
construction of the Baxter Q-operator family. Here, we reconstruct the Baxter
Q-operator and the same characterization of the spectrum by only using the SOV
method. This analysis allows us to deduce the main features required for the
extension to cyclic representations of other integrable quantum models of this
kind of spectrum characterization.
