On the number of eigenvalues of a model operator associated to a system of three-particles on lattices

A model operator H associated to a system of three particles on the three-dimensional lattice Z(3) that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators h(mu alpha)(0), a = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators h(mu alpha)(0), a = 1, 2, has a threshold eigenvalue.