On the essential spectrum of a model operator associated with the system of three particles on a lattice

A model operator $H$ associated with the system of three-identical particles on a lattice $\mathbb{Z}^3$ is considered. The location of the essential spectrum of $H$ is described by the spectrum of the corresponding Friedrichs model, that is, the two-particle and three-particle branches of the essential spectrum of $H$ are singled out. It is proved that the essential spectrum of $H$ consists of no more than three bounded closed intervals. An appearance of two-particle branches on the both sides of the three-particle branch is shown. Moreover, we obtain an analogue of the Faddeev equation and its symmetric version, for the eigenfunctions of $H.$