Asymptotics for the number of eigenvalues of three-particle Schrödinger operators on lattices

Abstract

We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schrödinger operator $H_γ(K),$ $K$ being the total quasi-momentum and $γ>0$ the ratio of the mass of fermion and boson. We choose for $γ>0$ the interaction $v(γ)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance. We prove for any $γ> 0$ the existence infinitely many eigenvalues of the operator $H_γ(0).$ We establish for the number $N(0,γ; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \lim_{z\to 0-}\frac{N(0,γ;z)}{\mid \log \mid z\mid \mid}={U} (γ) .$$ Moreover, for all nonzero values of the quasi-momentum $K \in T^3 $ we establish the finiteness of the number $ N(K,γ;τ_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,γ;0)$ of eigenvalues below zero.