Efimov effect in an analytically solvable model

The Efimov effect is demonstrated in a model consisting of two heavy particles and a light one when the light-heavy interaction leads to a zero-energy two-body bound state. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable S-wave potential of Yamaguchi form. It is demonstrated that in the case of a- two-body zero-energy bound state the binding energy of the light particle in the two-center potential exactly yields an effective $\text{1}\text{r}{}^2$ potential for the relative motion of the heavies. If the light-heavy mass ratio is made small enough, infinitely many bound states (the Efimov effect) are obtained. The approach to this limit is studied and the nature of the potential for large scattering length is obtained. An upper bound for the number of bound states is calculated using a result of Bargmann and Calogero and Efimov's $\text{ln}\text{(}\text{a}\text{r}{}_0\text{)}$ result is found. We note that the long-range effect arises from the large extent of the bound state, the pair wave function being essentially $\text{exp}\text{(?}\text{r}\text{a}\text{)}$ when the scattering length a is large.