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.

Key words Discrete Schrödinger operators, quantum mechanical two-and three-particle systems, Hamiltonians, short-range potentials, eigenvalues, quasi-momentum, essential spectrum, lattice, Faddeev type equation MSC (2000) Primary: 81Q10, Secondary: 35P20, 47N50A system of three quantum particles on the three-dimensional lattice Z 3 with arbitrary dispersion functions having not necessarily compact support and interacting via short-range pair potentials is considered. The energy operators of the systems of the two-and three-particles on the lattice Z 3 in the coordinate and momentum representations are described as bounded self-adjoint operators on the corresponding Hilbert spaces. For all sufficiently small values of the two-particle quasi-momentum k ∈ (−π, π] 3 the finiteness of the number of eigenvalues of the two-particle discrete Schrödinger operator hα(k) below the continuous spectrum is established. The location of the essential spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ (−π, π] 3 being the three-particle quasi-momentum, is described by means of the spectrum of the two-particle discrete Schrödinger operator hα(k), k ∈ (−π, π] 3 . It is established that the essential spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ (−π, π] 3 , consists of finitely many bounded closed intervals.

ABSTRACT.A model operator H associated to a system of three-particles on the three dimensional lattice Z 3 and interacting via pair non-local potentials is studied. The following results are proven: (i) the operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point, in the case, where both Friedrichs model operators hµ α (0), α = 1, 2, have threshold resonances. (ii) the operator H has a finite number of eigenvalues lying outside of the essential spectrum, in the case, where at least one of hµ α (0), α = 1, 2, has a threshold eigenvalue.

A family of Friedrichs models under rank one perturbations h μ (p), p ∈ (−π, π] 3 , μ > 0, associated to a system of two particles on the three-dimensional lattice Z 3 is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h μ (p) for all non-trivial values of p under the assumption that h μ (0) has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.