Zero-Threshold Resolvent Asymptotics of Three-Body Schroedinger Operators

We analyze the spectral properties for three-body Schroedinger operators at the threshold zero and give some results on the asymptotics of resolvent under the condition that zero is a regular point for all two-body subhamiltonians. To study the Efimov effect for N-body Schroedinger operators when the unique three-cluster assumption is not satisfied, the first step is to understand the singularities at the threshold of the resolvent of three-cluster subhamiltonians. In this work, we analyze the threshold zero for three-body Schroedinger operators under the assumption that the potentials are weak enough so that none of the two-body subhamiltonians has negative eigenvalues, nor zero resonance. It reveals that the related mathematical questions are highly nontrivial. We discuss the notion of three-body zero resonance and give the asymptotics of the resolvent when zero is a regular point. We also give a brief discuss on some open questions on this topic. (author)