The leading asymptotic terms of the three-body Coulomb scattering wave function

The asymptotic wave function derived by Alt and Mukhamedzhanov [Phys. Rev. A 47, 2004 (1993)] and Mukhamedzhanov and Lieber [Phys. Rev. A 54, 3078 (1996)] has been refined in the region where the pair $(\beta ,\gamma )$ remains close to each other while the third particle $\alpha$ is far away from them $({\rho }_{\alpha }\to \mathrm{\infty }$, $r_{\alpha }\mathrm{/}{\rho }_{\alpha }\to 0)$. The improved wave function satisfies the Schr"odinger equation up to the terms of order $O(1\mathrm{/}{{\rho }_{\alpha }}^3)$, provides the leading asymptotic terms of the three-body scattering wave function with Coulomb interactions and gives further insight into the continuum behavior of the three-charged-particle wave function and helps to obtain $3\to 3$ scattered wave. This opens up further ways of solving and analysing the three-body Schr"odinger equation by numerical means.