Branches of the essential spectrum of the lattice spin-boson model with at most two photons

We consider a lattice analogue of the Am model of light radiation with a fixed atom and at most m photons (m = 1, 2). We describe the essential spectrum of the operator A2 in terms of the spectrum of the operator A1, i.e., we find the “two-particle” and “three-particle” branches of the essential spectrum of A2. We prove that the essential spectrum is a union of at most six intervals, and we study their positions. We derive an estimate for the lower bound of the “two-particle” and “three-particle” branches.