Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-π; π]^3$ 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 $Λ\subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in Λ$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in Λ,$ the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0

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