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

RESUMO

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 $|x_N|\to\infty$, $\psi(x_1,\dots,x_N)/\sqrt{\rho(x_N)}= \phi(x_1,\dots,x_{N-1})+O(|x_N|^{-2+\alpha} \ln |x_N|)$ uniformly in the region $|x_i|\le |x_N|^{\alpha}$. In the region $|x_i|\le R_0\ln |x_N|$, the convergence rate is $O(|x_N|^{-2} (\ln |x_N|)^2)$. We also establish an $O(|x_N|^{-2})$ bound on the squared $L^2$ error. These results hold both below the essential spectrum and at threshold.