Methods of Functional Analysis and Topology

Аннотация

Abstract. In the present paper we consider a family of 2 × 2 operator matrices Aµ(k), k ∈ T3:= (−π, π]3, µ > 0, associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z3, interacting via creation and annihilation operators. We prove that there is a value µ0 of the parameter µ such that only for µ = µ0 the operator Aµ(0) has a virtual level at the point z = 0 = min σess(Aµ(0)) and the operator Aµ(π) has a virtual level at the point z = 18 = max σess(Aµ(π)), where 0 := (0, 0, 0), π := (π, π, π) ∈ T3. Theabsence of the eigenvalues of Aµ(k) for all values of k under the assumption that µ = µ0 is shown. The threshold energy expansions for the Fredholm determinant associated to Aµ(k) are obtained