The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice $\mathbb{Z}^3$ and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator $H(K)$, where $K$ is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of $H(K)$ for all sufficiently small values of the zero-range attractive potentials is established.
The asymptotics $\lim_{z\to 0-}\frac{N(0,z)}{|\!\log|z||}=\mathcal{U}_0$ is found for the number of eigenvalues $N(0,z)$ lying below $z<0$. Moreover, for all sufficiently small nonzero values of the three-particle quasimomentum $K$, the finiteness of the number $N(K,\tau_{\operatorname{ess}}(K))$ of eigenvalues below the essential spectrum of $H(K)$ is established and the asymptotics of the number $N(K,0)$ of eigenvalues of $H(K)$ below zero is given.