Threshold analysis for a family of $2\times2$ operator matrices

We consider a family of $2\times2$ operator matrices $\mathcal{A}_\mu(k)$, $k\in\mathbb{T}^3:=(-\pi;\pi]^3$, $\mu>0$, 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 $\mathbb{Z}^3$, interacting via annihilation and creation operators. We find a set $\Lambda:=\{k^{(1)},\dots,k^{(8)}\}\subset\mathbb{T}^3$ and a critical value of the coupling constant $\mu$ to establish necessary and sufficient conditions for either $z=0=\min\limits_{k\in\mathbb{T}^3}\sigma_{\mathrm{ess}}(\mathcal{A}_\mu(k))$ (or $z=27/2=\max\limits_{k\in\mathbb{T}^3}\sigma_{\mathrm{ess}}(\mathcal{A}_\mu(k))$) is a threshold eigenvalue or a virtual level of $\mathcal{A}_\mu(k^{(i)})$ for some $k^{(i)}\in\Lambda$.