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The analysis of the transfer matrices associated to the most generalrepresentations of the 8-vertex reflection algebra on spin-1/2 chains is hereimplemented by introducing a quantum separation of variables (SOV) method whichgeneralizes to these integrable quantum models the method first introduced bySklyanin. More in detail, for the representations reproducing in theirhomogeneous limits the open XYZ spin-1/2 quantum chains with the most generalintegrable boundary conditions, we explicitly construct representations of the8-vertex reflection algebras for which the transfer matrix spectral problem isseparated. Then, in these SOV representations we get the completecharacterization of the transfer matrix spectrum (eigenvalues and eigenstates)and its non-degeneracy. Moreover, we present the first fundamental step towardthe characterization of the dynamics of these models by deriving determinantformulae for the matrix elements of the identity on separated states, whichapply in particular to transfer matrix eigenstates. The comparison of ouranalysis for the 8-vertex reflection algebra with that of [1, 2] for the6-vertex one leads to the interesting remark that a profound similarity in boththe characterization of the spectral problems and of the scalar products existsfor these two different realizations of the reflection algebra once they aredescribed by SOV method. As it will be shown in a future publication, thisremarkable similarity will be at the basis of the simultaneous determination ofform factors of local operators of integrable quantum models associated togeneral reflection algebra representations of both 8-vertex and 6-vertex type.

SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra