Eigenvectors of the Baxter–Bazhanov–Stroganov $\tau^{(2)}(t_q)$ model with fixed-spin boundary conditions

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter–Bazhanov–Stroganov {(}BBS{\rm)} model {\rm(}$N$-state spin model{)} with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin's method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding $T$–$Q$ Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like $\mathbb{Z}_N$ quantum chain model.