Existence Condition for the Eigenvalue of a Three-Particle Schrödinger Operator on a Lattice

Abstract

A three-particle discrete Schrödinger operator ${{H}_{{\mu ,\gamma }}}({\mathbf{K}})$, ${\mathbf{K}} \in {{\mathbb{T}}^{3}}$ associated with a system of three particles (two fermions with the mass 1 and one more particle with the mass $m = {\text{1/}}\gamma < 1$) interacting through pairwise repulsive zero-range potentials $\mu > 0$ on the three-dimensional lattice ${{\mathbb{Z}}^{3}}$ is considered. The operator ${{H}_{{\mu ,\gamma }}}({\boldsymbol{\pi }})$, ${\boldsymbol{\pi }} = (\pi ,\pi ,\pi )$ is proved to have no eigenvalues for $\gamma \in (1,{{\gamma }_{0}})$ (${{\gamma }_{0}} \approx 4.7655$) and have the unique eigenvalue with multiplicity three for $\gamma > {{\gamma }_{0}}$, which lies to the right of the essential spectrum for sufficiently big $\mu$.