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

Abstract

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 $\mathop {\lim }\limits_{z \to 0 - } \frac{{N(0,z)}}{{|\log |z||}} = U_{\text{0}}^{}$ 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 $N(K,\tau _{ess} (K))$ 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.