Asymptotics of Ground States for Helium-Like and Bosonic Atoms in a Two-Cluster Region

We study the asymptotic behavior of positive ground states of $N$-particle atomic Coulomb Hamiltonians in a two-cluster region. For the $N$-particle ground state $\psi$, its one-particle density $\rho$, and $(N-1)$-particle ground state $\phi$, we show that for $0<\alpha <1$, as $\mathrm{\mid }{x}_N\mathrm{\mid }\to \mathrm{\infty }$, $\psi (x_1,\dots ,x_N)\mathrm{/}\sqrt{\rho (x_N)}=\phi (x_1,\dots ,x_{N-1})+O(\mathrm{\mid }{x}_N{\mathrm{\mid }}^{-2+\alpha }\ln \mathrm{\mid }{x}_N\mathrm{\mid })$ uniformly in the region $\mathrm{\mid }{x}_i\mathrm{\mid }\le \mathrm{\mid }{x}_N{\mathrm{\mid }}^{\alpha }$. In the region $\mathrm{\mid }{x}_i\mathrm{\mid }\le R_0\ln \mathrm{\mid }{x}_N\mathrm{\mid }$, the convergence rate is $O(\mathrm{\mid }{x}_N{\mathrm{\mid }}^{-2}(\ln \mathrm{\mid }{x}_N\mathrm{\mid }{)}^2)$. We also establish an $O(\mathrm{\mid }{x}_N{\mathrm{\mid }}^{-2})$ bound on the squared $L^2$ error. These results hold both below the essential spectrum and at threshold.