We consider the family of 3 × 3 operator matrices H(K), K ∈ T d := (−π; π] d arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus T d . We obtain an analogue of the Faddeev equation for the eigenfunctions of H(K). An analytic description of the essential spectrum of H(K) is established. Further, it is shown that the essential spectrum of H(K) consists the union of at most three bounded closed intervals.

We consider the family of 3 × 3 operator matrices H(K), K ∈ T 3 := (−π; π] 3 associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set Λ ⊂ T 3 to prove the existence of infinitely many eigenvalues of H(K) for all K ∈ Λ when the associated Friedrichs model has a zero energy resonance. It is found that for every K ∈ Λ, the number N (K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N (K, z)| log |z|| −1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ. Moreover, we prove that for any K ∈ Λ the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.