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The integrable quantum models, associated to the transfer matrices of the6-vertex reflection algebra for spin 1/2 representations, are studied in thispaper. In the framework of Sklyanin's quantum separation of variables (SOV), weprovide the complete characterization of the eigenvalues and eigenstates of thetransfer matrix and the proof of the simplicity of the transfer matrixspectrum. Moreover, we use these integrable quantum models as further keyexamples for which to develop a method in the SOV framework to compute matrixelements of local operators. This method has been introduced first in [1] andthen used also in [2], it is based on the resolution of the quantum inverseproblem (i.e. the reconstruction of all local operators in terms of the quantumseparate variables) plus the computation of the action of separate covectors onseparate vectors. In particular, for these integrable quantum models, which inthe homogeneous limit reproduce the open spin-1/2 XXZ quantum chains withnon-diagonal boundary conditions, we have obtained the SOV-reconstructions fora class of quasi-local operators and determinant formulae for thecovector-vector actions. As consequence of these findings we provide onedeterminant formulae for the matrix elements of this class of reconstructedquasi-local operators on transfer matrix eigenstates.

Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators