Infinite number of eigenvalues of $2\times 2$ operator matrices: Asymptotic discrete spectrum

We study an unbounded $2\times2$ operator matrix $\mathcal{A}$ in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of $\mathcal{A}$. We consider a $2\times2$ operator matrix $\mathcal{A}_\mu$, where $\mu>0$ is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice $\mathbb{Z}^3$. We find the critical value $\mu_0$ of the coupling constant $\mu$ for which $\mathcal{A}_{\mu_0}$ has an infinite number of eigenvalues. These eigenvalues accumulate at the lower and upper bounds of the essential spectrum. We obtain an asymptotic formula for the number of such eigenvalues in both the left and right parts of the essential spectrum.