2×2 operator matrix with real parameter and its spectrum

¹ Bukhara State University, 200100 Bukhara, Uzbekistan
² Bukhara State Pedagogical Institute, Bukhara City, Piridastgir, 2, 200100, Uzbekistan
³ Navoi State Pedagogical Institute, 210100 Navoi, Uzbekistan

^* Corresponding author: elyor.dilmurodov@mail.ru, e.b.dilmurodov@buxdu.uz

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.