Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

International audienceWe study the transfer matrix spectral problem for the cyclic representationsof the trigonometric 6-vertex reflection algebra associated to theBazhanov-Stroganov Lax operator. The results apply as well to the spectralanalysis of the lattice sine-Gordon model with integrable open boundaryconditions. This spectral analysis is developed by implementing the method ofseparation of variables (SoV). The transfer matrix spectrum (both eigenvaluesand eigenstates) is completely characterized in terms of the set of solutionsto a discrete system of polynomial equations in a given class of functions.Moreover, we prove an equivalent characterization as the set of solutions to aBaxter's like T-Q functional equation and rewrite the transfer matrixeigenstates in an algebraic Bethe ansatz form. In order to explain our methodin a simple case, the present paper is restricted to representations containingone constraint on the boundary parameters and on the parameters of theBazhanov-Stroganov Lax operator. In a next article, some more technical tools(like Baxter's gauge transformations) will be introduced to extend our approachto general integrable boundary conditions