Analysis of the spectrum of a $2\times 2$ operator matrix. Discrete spectrum asymptotics

We consider a $2 \times 2$ operator matrix ${\mathcal A}_\mu,$ $\mu>0$ 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 analogue of the Faddeev equation and its symmetric version for the eigenfunctions of ${\mathcal A}_\mu$. We describe the new branches of the essential spectrum of ${\mathcal A}_\mu$ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of ${\mathcal A}_\mu$ consists the union of at most three bounded closed intervals and their location is studied. For the critical value $\mu_0$ of the coupling constant $\mu$ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of ${\mathcal A}_\mu$. In this case, an asymptotic formula for the discrete spectrum of ${\mathcal A}_\mu$ is found.