Discrete spectrum asymptotics for the three-particle Hamiltonians on lattices

We consider the Hamiltonian of a system of three quantum mechanical particles on the three-dimensional lattice $\Z^3$ interacting via short-range pair potentials. We prove for the two-particle energy operator $h(k),$ $k\in \T^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(k)$ lying below the essential spectrum under assumption that the operator $h(0)$ corresponding to the zero value of $k$ has a zero energy resonance. We describe the location of the essential spectrum of the three-particle discrete Schr\"{o}dinger operators $H(K)$,$K$ the three-particle quasi-momentum by the spectra of $h(k), k\in \T^3.$ We prove the existence of infinitely many eigenvalues of H(0) and establish for the number of eigenvalues $N(0,z)$ lying below $z<0$ the asymptotics \begin{equation*}\label{asimz} \lim\limits_{z \to -0}\frac{N(0,z)}{|\log |z||}=\frac{\lambda_0}{2\pi}, \end{equation*} where $\lambda_0$ a unique positive solution of the equation $$\lambda = \frac{8 \sinh \pi\lambda /6}{\sqrt 3 \cosh \pi\lambda/2}.$$ We prove that for all $K \in U_\delta^0(0),$ where $U_\delta^0(0)$ some punctured $\delta >0$ neighborhood of the origin, the number $N(K,0)$ of eigenvalues the operator $H(K)$ below zero is finite and satisfy the asymptotics \begin{equation*}\label{asimk} \lim\limits_{|K| \to 0}\frac{N(K,0)}{|\log |K||}=\frac{\lambda_0}{\pi}. \end{equation*}