Eigenvalues and virtual levels of a family of $2 \times 2$ operator matrices

In the present paper we consider a family of $2 \times 2$ operator matrices ${\mathcal A}_μ(k),$ $k \in {\Bbb T}^3:=(-π, π]^3,$ $μ>0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\Bbb Z}^3,$ interacting via creation and annihilation operators. We prove that there is a value $μ_0$ of the parameter $μ$ such that only for $μ=μ_0$ the operator ${\mathcal A}_μ(\overline{0})$ has a virtual level at the point $z=0=\minσ_{\rm ess}({\mathcal A}_μ(\overline{0}))$ and the operator ${\mathcal A}_μ(\overlineπ)$ has a virtual level at the point $z=18=\maxσ_{\rm ess}({\mathcal A}_μ(\overlineπ))$, where $\overline{0}:=(0,0,0), \overlineπ:=(π,π,π) \in {\Bbb T}^3.$ The absence of the eigenvalues of ${\mathcal A}_μ(k)$ for all values of $k$ under the assumption that $μ=μ_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal A}_μ(k)$ are obtained.

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