On the number of eigenvalues of a model operator associated to a system of three-particles on lattices

Abstract

A model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, has a threshold eigenvalue.