We analyse the famous Baxter's T −Q equations for XXX (XXZ) spin chain and show that apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe-Ansatz equations, there exists also the second solution which should corresponds to Bethe-Ansatz beyond N/2. This second solution of Baxter's equation plays essential role and together with the first one gives rise to all fusion relations.

The Friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes. Such eigenvectors are called Gamow vectors and they acquire meaning in extensions of the conventional Hilbert space of quantum theory to the so-called rigged Hilbert space. In this paper, Gamow vectors are constructed for a solvable model of an unstable relativistic field. As a result, we obtain a time asymmetric relativistic extension of the Fock space. This extension leads to two distinct Poincaré semigroups. The time reversal transformation maps one semigroup to the other. As a result, the usual PCT invariance should be extended. We show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity.

The full set of polynomial solutions of the nested Bethe Ansatz is constructed for the case of A 2 rational spin chain. The structure and properties of these associated solutions are more various then in the case of usual XXX (A 1 ) spin chain but their role is similar.