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 $ψ$, 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)