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 three dimensional lattice ${\mathbb{Z}}^3$ 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 $h_{{\mu }_{\alpha }}(0),\alpha =1,2,$ 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 $h_{{\mu }_{\alpha }}(0),\alpha =1,2,$ has a threshold eigenvalue.

14 pages