"Schrödinger inequalities" and asymptotic behavior of many-electron densities

Upper bounds are derived to the many-electron density ρₖ(x₁, , xₖ) (2≤k≤n) of an n-electron atomic or molecular wave function. The derivation is based on former results on the decay of the one-electron density and on the "Schr\"odinger inequality" for (ρₖ)1/2 which, by means of a comparison theorem, allows the deduction of upper bounds for ρₖ in the region Gᵏ=(x₁, , xₖ)|rᵢ≥c, 1≤i≤k, c being a constant and rᵢ=|xᵢ|, provided an upper bound to ρₖ is available on the boundary ∂Gᵏ. The latter can be obtained from bounds to ρ_k-1. A recurrence procedure leads to the final result which for the case of an atom is given by ${{ρ}}ₖ{≤}dS{Π}{i=1}ᵏ[{r}ᵢ^{2[Z/2{(2{{ε}}ᵢ)}1/2{-}1]} {e}^{{-}{(2{{ε}}ᵢ)}1/2}{r}ᵢ]$ ($d$ is a constant, $S$ acts as a symmetrizer, $Z$ denotes the nuclear charge, and the ${{ε}}ᵢ$ ($1{≤}i{≤}k$) denote the successive ionization potentials of the state under consideration). We further report an improved bound for ${{ρ}}₂$ which depends explicitly on the interelectronic distance.