On the number of eigenvalues of the family of operator matrices

We consider the family of operator matrices $H(K)$, $K\in\mathbb{T}^3:=(-\pi,\pi]^3$ acting in the direct sum of zero-, one- and two-particle subspaces of the bosonic Fock space. We find a finite set $\Lambda\subset\mathbb{T}^3$ to establish the existence of infinitely many eigenvalues of $H(K)$ for all $K\in\Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K\in\Lambda$ the number $N(K,z)$ of eigenvalues of $H(K)$ lying on the left of $z$, $z<0$, satisfies the asymptotic relation $\lim_{z\to -0}N(k,z)|\log|z||^{-1}=\mathcal{U}_0$ with $0<\mathcal{U}_0<\infty$, independently on the cardinality of $\Lambda$. Moreover, we show that for any $K\in\Lambda$ 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.