The number of eigenvalues of three-particle Schroedinger operators on lattices

We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and a boson) on the three-dimensional lattice Z3 interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schroedinger operator Hγ(K), K being the total quasi-momentum belonging to the three-dimensional torus T3=(-π,π]3 and γ > 0 the ratio of the mass of fermion to boson. We choose for γ > 0 the interaction μ(γ) in such a way that the system consisting of one fermion and one boson has a zero-energy resonance. For all nonzero values of the quasi-momentum K element of T3 , we prove the finiteness of the number N(K, γ; τγ(K)) of eigenvalues of Hγ(K) below the bottom τγ(K) of the essential spectrum and we give for N(K, γ; 0) an asymptotics as K → 0. Moreover, we prove the existence of infinitely many eigenvalues of the operator Hγ(0) and give for the number N(0, γ; z) of eigenvalues lying below z < 0 an asymptotics as z → 0