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 $ψ$, its one-particle density $ρ$, and $(N-1)$-particle ground state $φ$, we show that for $0<α<1$, as $|x_N|\to\infty$, $$ψ(x_1,\dots,x_N)/\sqrt{ρ(x_N)}= φ(x_1,\dots,x_{N-1})+O(|x_N|^{-2+α} \ln |x_N|)$$ uniformly in the region $|x_i|\le |x_N|<sup>α$. In the region $|x_i|\le R_0\ln |x_N|$, the convergence rate is $O(|x_N|^{-2} (\ln |x_N|)<sup>2)$. We also establish an $O(|x_N|<sup>{-2})$ bound on the squared $L<sup>2$ error. These results hold both below the essential spectrum and at threshold.