The essential and discrete spectrum of a model operator associated to a system of three identical quantum particles

A model operator H associated to a system of three identical quantum particles on the three-dimensional lattice ℤ3 is considered. The existence of eigenvalues lying below the essential spectrum of a family of Friedrichs models under rank-one perturbations hμα(p), p∈T3 , α = 1, 2, is established. The essential spectrum of the operator H is described by the spectrum of the family of the Friedrichs models hμα(p), p∈T3 , α = 1, 2. The following results are proven: The operator H has a finite number of eigenvalues lying below zero, if at least one of the Friedrichs models hμα(0), α = 1, 2, has a zero energy resonance. The operator H has infinitely many eigenvalues lying below zero and accumulating at zero, if both operators hμα(0), α = 1,2, have zero energy resonances.