Minlos-Faddeev regularization of zero-range interactions in the three-body problem

To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter $\sigma$. Explicitly, Efimov or Thomas effects remain for $\sigma < \sigma_c$, the additional boundary conditions of two types should be imposed at the triple-collision point for $\sigma_c \le \sigma < \sigma_e$ and $\sigma_e < \sigma < \sigma_r$, and the problem is regularized for $\sigma \ge \sigma_r$. Critical values $\sigma_c < \sigma_e < \sigma_r$ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.