Decay properties of zero-energy resonances of multi-particle Schr\"odinger operators and why the Efimov effect does not exist for systems of $N\geq 4$ particles.

We consider $N$-body Schr\"odinger operators with a virtual level at the threshold of the essential spectrum. We show that in the case of $N\geq 3$ particles in dimension $n\geq3$ virtual levels correspond to eigenvalues of the system and we obtain decay rates of the corresponding eigenfunctions in dependence on the dimension and the number of particles. We prove that in dimension $n\geq 3$ the Hamiltonian of $N\geq 4$ particles interacting via short-range potentials admits only a finite number of negative eigenvalues. We extend our results to dimension $n=1$ and $n=2$ in case of $N\geq 4$ fermions.