Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory

The transfer matrix of the XXZ open spin-1/2 chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q|=1) is diagonalized using the representation theory of the q-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d=2^N. The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results by Nepomechie et al. and Cao et al.. In the latter case the complete family of eigenvalues and eigenstates splits in two sets, each associated with a characteristic polynomial of degree $d< 2^N$. Numerical checks performed for small values of $N$ support the analysis.