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

Abstract

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 $\varphi$, we show that for $0<\alpha <1$, as $|x_N|\to \infty$, $\psi (x_1,\dots ,x_N)/\sqrt{\rho (x_N)}=\varphi (x_1,\dots ,x_{N-1})+O(|x_N{|}^{-2+\alpha }\ln |x_N|)$ 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.

17 pages