2×2 operator matrix with real parameter and its spectrum

In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix Aμ (so called generalized Friedrichs model) with real parameter μ ∈ R. It is associated with the Hamiltonian of a system consisting of at most two particles on a d -dimensional lattice Zd, interacting via creation and annihilation operators. Aμ is linear bounded self-adjoint operator acting in the two-particle cut subspace of the Fock space, that is, in the direct sum of zero-particle and one-particle subspaces of a Fock space. We find the essential and discrete spectra of the block operator matrix Aμ. The Fredholm determinant and resolvent operator associated to Aμ are constructed. The spectrum of Aμ plays an important role in the study of the spectral properties of the Hamiltonians associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles on a lattice.