Existence Condition for the Eigenvalue of a Three-Particle Schrödinger Operator on a Lattice
Аннотация
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 $$\mu $$ .