The first Schur complement for a lattice spin-boson model with at most two photons

In the present paper, we consider a lattice spin-boson model $\mathcal{A}_2$ with a fixed atom and at most two photons. We construct the first Schur complement $S_1(\lambda)$ with spectral parameter $\lambda$ corresponding to $\mathcal{A}_2$. We prove the Birman–Schwinger principle for $\mathcal{A}_2$ with respect to $S_1(\lambda)$. We investigate an important properties of $S_1(\lambda)$ related to the number of eigenvalues of $\mathcal{A}_2$ for all dimensions d of the torus $\mathbb{T}^{\mathrm{d}}$ and for any coupling constant $\alpha>0$.