Interaction range perturbation theory for three-particle problem

The limit of zero interaction range is correctly defined for a system of three spinless particles and three particles in a doublet state. The scattering amplitude is expanded with respect to the interaction range $r$, and the corrections of order $r\ln r$, $r$, and $r^2\ln^2r$ are found. An explicit model-independent asymptotic expression is obtained for the scattering amplitude in terms of the scattering length, and its accuracy is established.