Universality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a lattice

In the present paper, we consider the Hamiltonian $H(K)$, $K\in\mathbb T^3:=(-\pi,\pi]^3$ of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set $\Lambda\subset \mathbb T^3$ such that for all values of the total quasi-momentum $K\in\Lambda$ the operator $H(K)$ has infinitely many negative eigenvalues accumulating at zero. It is found that for every $K\in\Lambda$, the number $N(K;z)$ of eigenvalues of $H(K)$ lying on the left of $z$, $z<0$, satisfies the asymptotic relation $\lim\limits_{z\to-0}N(K;z)\bigl|\log|z|\bigr|^{-1}=\mathcal U_0$ with $0<\mathcal U_0<\infty$, independently on the cardinality of $\Lambda$.