  The spectral properties of operators formed from generators of the q-Onsager
non-Abelian infinite dimensional algebra are investigated. Using a suitable
functional representation, all eigenfunctions are shown to obey a second-order
q-difference equation (or its degenerate discrete version). In the algebraic
sector associated with polynomial eigenfunctions (or their discrete analogues),
Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz
approach is not applicable in related massive quantum integrable models, the
eigenfunctions are also described. The spin-half XXZ open spin chain with
general integrable boundary conditions is reconsidered in light of this
approach: all the eigenstates are constructed. In the algebraic sector which
corresponds to special relations among the parameters, known results are
recovered.
