We consider a family of 2 × 2 operator matrices Aµ(k), k ∈ T 3 := (−π, π] 3 , µ > 0, acting in the direct sum of zero-and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z 3 , interacting via annihilation and creation operators. We find a set Λ := {k (1) , ..., k (8) } ⊂ T 3 and a critical value of the coupling constant µ to establish necessary and sufficient conditions for either z = 0 = min k∈T 3 σess(Aµ(k)) ( or z = 27/2 = max k∈T 3 σess(Aµ(k)) is a threshold eigenvalue or a virtual level of Aµ(k (i) ) for some k (i) ∈ Λ.

We consider a 2 × 2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ. We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied. For the critical value µ 0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ. In this case, an asymptotic formula for the discrete spectrum of Aµ is found.

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We consider a 2 × 2 operator matrix Aμ, μ > 0, related with the lattice systems describing three particles in interaction, without conservation of the number of particles on a d-dimensional lattice. We obtain an analogue of the Faddeev type integral equation for the eigenfunctions of Aμ. We describe the two- and three-particle branches of the essential spectrum of Aμ via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of Aμ consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of Aμ with respect to the dimension d ∈ N of the torus Td and the coupling constant μ > 0.

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