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

We consider a lattice analogue of the $\mathcal A_m$ model of light radiation with a fixed atom and at most $m$ photons $(m=1,2)$. We describe the essential spectrum of the operator $\mathcal A_2$ in terms of the spectrum of the operator $\mathcal A_1$, i.e., we find the “two-particle” and “three-particle” branches of the essential spectrum of $\mathcal A_2$. 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.