Finiteness of discrete spectrum of three particle Schrödinger operator on the lattice

A Hamiltonian system of three identical quantum particles on a lattice interacting via pairwise contact attracting potentials is discussed. Finiteness of three particle bound states of the three dimensional Schrödinger operator is proved under the condition that operators describing two particle subsystems do not have virtual levels. For high dimensions $(\nu\geq5)$ the finiteness of three particle bound states is also proved under the presence of virtual levels.