Essential and discrete spectrum of a three-particle lattice Hamiltonian with non-local potentials

We consider a model operator (Hamiltonian) $H$ associated with a system of three particles on a d-dimensional lattice that interact via non-local potentials. Here the kernel of non-local interaction operators has rank $n$ with $n\ge 3$. We obtain an analog of the Faddeev equation for the eigenfunctions of $H$ and describe the spectrum of $H$. It is shown that the essential spectrum of H consists the union of at most $n+1$ bounded closed intervals. We estimate the lower bound of the essential spectrum of $H$ for the case d = 1.