Discrete Spectrum of a Model Operator Related to Three-Particle Discrete Schr\"{o}dinger Operators

Abstract

A model operator $H_\mu,$ $\mu>0$ associated to a system of three particles on the three-dimensional lattice $ \mathbb{Z}^3$ that interact via nonlocal pair potentials is considered. We study the case where the parameter function $w$ has a special form with the non degenerate minimum at the $n, n>1$ points of the six-dimensional torus $\mathbb{T}^6.$ If the associated Friedrichs model has a zero energy resonance, then we prove that the operator $H_\mu$ has infinitely many negative eigenvalues accumulating at zero and we obtain an asymptotics for the number of eigenvalues of $H_\mu$ lying below $z,$ $z<0$ as $z\to -0.$