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

We consider a family of $2 \times 2$ operator matrices ${\mathcal A}_μ(k),$ $k \in {\Bbb T}^3:=(-π, π]^3,$ $μ>0$, 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 ${\Bbb Z}^3,$ interacting via annihilation and creation operators. We find a set $Λ:=\{k^{(1)},...,k^{(8)}\} \subset {\Bbb T}^3$ and a critical value of the coupling constant $μ$ to establish necessary and sufficient conditions for either $z=0=\min\limits_{k\in {\Bbb T}^3} σ_{\rm ess}({\mathcal A}_μ(k))$ ( or $z=27/2=\max\limits_{k\in {\Bbb T}^3} σ_{\rm ess}({\mathcal A}_μ(k))$ is a threshold eigenvalue or a virtual level of ${\mathcal A}_μ(k^{(i)})$ for some $k^{(i)} \in Λ.$

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