Discrete spectrum asymptotics for the three-particle Hamiltonians on lattices

Abstract

We consider the Hamiltonian of a system of three quantum mechanical particles on the three-dimensional lattice ${\mathbb{Z}}^3$ interacting via short-range pair potentials. We prove for the two-particle energy operator $h(k),$ $k\in \T^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(k)$ lying below the essential spectrum under assumption that the operator $h(0)$ corresponding to the zero value of $k$ has a zero energy resonance. We describe the location of the essential spectrum of the three-particle discrete Schrödinger operators $H(K)$,$K$ the three-particle quasi-momentum by the spectra of $h(k), k\in \T^3.$ We prove the existence of infinitely many eigenvalues of H(0) and establish for the number of eigenvalues $N(0,z)$ lying below $z<0$ the asymptotics \begin{equation*}\label{asimz} \lim\limits_{z \to -0}\frac{N(0,z)}{|\log |z||}=\frac{λ_0}{2π}, \end{equation*} where ${\lambda }_0$ a unique positive solution of the equation $$\lambda =\frac{8\sinh \pi \lambda /6}{\sqrt{3}\cosh \pi \lambda /2}.$$ We prove that for all $K\in U_{\delta }^0(0),$ where $U_{\delta }^0(0)$ some punctured $\delta >0$ neighborhood of the origin, the number $N(K,0)$ of eigenvalues the operator $H(K)$ below zero is finite and satisfy the asymptotics \begin{equation*}\label{asimk} \lim\limits_{|K| \to 0}\frac{N(K,0)}{|\log |K||}=\frac{λ_0}π. \end{equation*}

25 pages