On the eigenvalues of the lattice spin-boson model with at most one photon

\nIn the present paper we consider a lattice spin-boson model with at most one photon A , which has a 2×2 block operator matrix representation. The essential spectrum of A is analyzed. We prove that the operator matrix A has four eigenvalues. We consider the case where the special integral is an infinite. The existence condition of the eigenvalues lying in and out of the essential spectrum are found. The results presented in this paper plays an important role when we study the location of the two-particle and three-particle branches of the essential spectrum of the lattice spin-boson Hamiltonian with at most two photons, and also to showing the finiteness of the number of its eigenvalues.\n