2×2 operator matrix with real parameter and its spectrum
Abstract
In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix A μ (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 Z d , interacting via creation and annihilation operators. A μ 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 A μ . The Fredholm determinant and resolvent operator associated to A μ are constructed. The spectrum of A μ 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.