Asymptotics of three-body bound state radial wave functions of halo nuclei involving two charged particles

Asymptotic expressions for the radial and full wave functions of a three{body bound halo nuclear system with two charged particles in relative coordinates are obtained in explicit form, when the relative distance between two particles tends to infinity. The obtained asymptotic forms are applied to the analysis of the asymptotic behavior of the three-body (pn?) wave functions for the halo ($E^*=3.562$ MeV, $J^{\pi}=0^+$, $T=1$) state of $^6$Li derived by D. Baye within the Lagrange-mesh method for two forms of the $\alpha N$ -potential. The agreement between the calculated wave function and the asymptotic formula is excellent for distances up to 30 fm. Information about the values of the three-body asymptotic normalization functions is extracted. It is shown that the extracted values of the three-body asymptotic normalization function are sensitive to the form of the $\alpha N$ -potential. The mirror symmetry is revealed for the three-body asymptotic normalization functions derived for the isobaric ($^6$He, $^6$Li) pair.