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 ${\mathbb{Z}}^3$ 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 $H_{\gamma }(K),$ $K$ being the total quasi-momentum and $\gamma >0$ the ratio of the mass of fermion and boson. We choose for $\gamma >0$ the interaction $v(\gamma )$ in such a way the system consisting of one fermion and one boson has a zero energy resonance. We prove for any $\gamma >0$ the existence infinitely many eigenvalues of the operator $H_{\gamma }(0).$ We establish for the number $N(0,\gamma ;z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$\underset{z\to 0-}{\lim }\frac{N(0,\gamma ;z)}{\mid \log \mid z\mid \mid }=U(\gamma ).$$ Moreover, for all nonzero values of the quasi-momentum $K\in T^3$ we establish the finiteness of the number $N(K,\gamma ;{\tau }_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\gamma ;0)$ of eigenvalues below zero.

25 pages