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

Upper bounds are derived to the many-electron density ${\rho }_k(x_1, \dots , x_k)$ ($2\le k\le n$) 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 "Schrödinger inequality" for ${\left({\rho }_k\right)}^{\frac{1}{2}}$ which, by means of a comparison theorem, allows the deduction of upper bounds for ${\rho }_k$ in the region $G^k=\left\{\right(x_1, \dots , x_k\left)\right|r_i\ge c, 1\le i\le k\}$, $c$ being a constant and $r_i=\left|x_i\right|$, provided an upper bound to ${\rho }_k$ is available on the boundary $\partial G^k$. The latter can be obtained from bounds to ${\rho }_{k-1\mathrm{}}$. A recurrence procedure leads to the final result which for the case of an atom is given by ${\rho }_k\le \mathrm{dS}\Pi {i=1\mathrm{}}^k\left[r_i^{2[\frac{Z}{2}{\left(2\mathrm{}{\epsilon }_i\right)}^{\frac{1}{2}}-1]} e^{-{\left(2\mathrm{}{\epsilon }_i\right)}^{\frac{1}{2}}}{r}_i\right]$ ($d$ is a constant, $S$ acts as a symmetrizer, $Z$ denotes the nuclear charge, and the ${\epsilon }_i$ ($1\le i\le k$) denote the successive ionization potentials of the state under consideration). We further report an improved bound for ${\rho }_2$ which depends explicitly on the interelectronic distance.