Spectrum of the energy operator of two-magnon systems in the isotropic Heisenberg ferromagnet model with impurity

We consider a two-magnon system in the isotropic Heisenberg ferromagnet model with impurity on a $\nu$-dimensional lattice $\mathbb Z^{\nu}$. We establish that the essential spectrum of the system consists of the union of at most four intervals. We obtain the lower and upper estimates for the number of three-particle bound states of the system.