Decay Rates of Bound States at the Spectral Threshold of Multi-Particle Schrödinger Operators

Abstract

We consider $N$-body Schrödinger operators with $N\ge 3$ quantum particles interacting via short-range potentials in dimension $d\ge 3$, where the essential spectrum coincides with the half line $[0,\infty )$. We give the asymptotic behaviour of eigenfunctions corresponding to the eigenvalue at the threshold of the essential spectrum under the condition that the eigenfunctions are not orthogonal to the sum of the pair interactions. This condition is fulfilled when zero is the smallest eigenvalue and the pair interactions are negative. We also give examples of systems when this condition is not met.