Threshold analysis for a family of 2 x 2 operator matrices

We consider a family of 2 x 2 operator matrices A(u)(k); k is an element of 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 Z(3), interacting via annihilation and creation operators. We find a set Lambda := {K-(1), ... ,K-(8)} subset of T-3 and a critical value of the coupling constant mu to establish necessary and sufficient conditions for either z = 0 = min(k is an element of T3) sigma(ess) (A(mu)(k)) (or z = 27/2 = max(k is an element of T3) sigma(ess)(A(mu)(k)) is a threshold eigenvalue or a virtual level of A(mu)(k((i))) for some k((i)) is an element of Lambda.