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

We consider a 2 x 2 operator matrix A(mu), mu > 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(mu). We describe the two- and three-particle branches of the essential spectrum of A(mu) via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of A(mu) consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of A(mu) with respect to the dimension d is an element of N of the torus T-d, and the coupling constant mu > 0.