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

Abstract

We consider a lattice analogue of the 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 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.