Eigenvalues and virtual levels of a family of 2×2 operator matrices - Rasulov, T. H.; Dilmurodov, E. B. (Methods of Functional Analysis and Topology, 2019)
In the present paper we consider a family of 2 × 2 operator matrices A_μ(k), k 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 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 A_μ(0) has a virtual level at the point z=0=σ_ess(A_μ(0)) and the operator A_μ(π) has a virtual level at the point z=18=σ_ess(A_μ(π)), where 0:=(0,0,0), π:=(π,π,π) T^3. The absence of the eigenvalues of 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 A_μ(k) are obtained.