Efimov's Effect: A New Pathology of Three-Particle Systems. II

By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length $a$ becomes large. [The number of bound states is then roughly $\left(\frac{1}{\pi }\right)\ln \left(\Lambda \mathrm{}\right|a\left|\right)$, where $\Lambda$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.