Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum

Abstract

We consider a system of $N$ pairwise interacting particles described by the Hamiltonian $H$, where $\sigma_{ess} (H) = [0,\infty)$ and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if $N \geq 4$ and none of the Hamiltonians corresponding to the subsystems containing $N-2$ or less particles has an eigenvalue equal to zero then $H$ has a finite number of negative energy bound states. This result provides a positive proof to a long--standing conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the "true" four--body Efimov effect.