The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice Z 3 and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with k ∈ T 3 = (−π, π] 3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k = 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), K ∈ T 3 being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N (0, z) of eigenvalues of H(0) lying below z < 0 the following limit existsMoreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number N (K, τess(K)) of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N (K, 0) of eigenvalues lying below zero is given.

For a wide class of two-body energy operators h(k) on the three-dimensional lattice Z 3 , k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k = 0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the twoparticle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the oneparticle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

ABSTRACT.A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The precise location and structure of the essential spectrum of H is described. The existence of infinitely many eigenvalues below the bottom of the essential spectrum of H is proved for the case where an associated generalized Friedrichs model has a resonance at the bottom of its essential spectrum. An asymptotics for the number N (z) of eigenvalues below the bottom of the essential spectrum is also established. The finiteness of eigenvalues of H below the bottom of the essential spectrum is proved if the associated generalized Friedrichs model has an eigenvalue with energy at the bottom of its essential spectrum.