On the finiteness of the discrete spectrum of four-particle lattice Schrِdinger operators

A Hamiltonian describing four quantum mechanical particles (bosons) moving on a lattice is considered. The corresponding Fredholmʹs integral equations of the Faddeev-Yakubovskii and Weinberg type are obtained and the location and structure of the essential spectrum are studied. The finiteness of the discrete spectrum for all interactions and the absence of eigenvalues lying outside the essential spectrum for the case of “weak interactions” are proved.