Estimates for the Bounds of the Essential Spectrum of a 2 × 2 Operator Matrix

Abstract

We consider a 2 × 2 operator matrix A_μ, μ > 0, related to the lattice systems describing three particles in interaction, without conservation of the number of particles on a d-dimensional lattice. We obtain an analogue of the Faddeev type integral equation for the eigenfunctions of A_μ. We describe the two- and three-particle branches of the essential spectrum of A_μ via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of A_μ consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of A_μ with respect to the dimension d ∈ N of the torus T^d, and the coupling constant μ > 0.