"Schrödinger inequalities" and asymptotic behavior of the electron density of atoms and molecules

Within the frame of the infinite-nuclear-mass approximation a differential inequality for the square root of the one-electron density is derived. This linear differential inequality is structured like a one-particle Schrödinger equation and leads to results on the analytic behavior of the electron density $\rho$ in the region far from the nuclei. A domain determined by the potential and the ionization energy is given where $\rho$ is subharmonic. For atoms it is shown that the spherically averaged electron density is a convex monotonically decreasing function outside some sphere whose radius depends on the ionization energy and the electron nuclear attraction. Furthermore, an upper bound for the electron density is given in terms of Whittaker functions which decreases exponentially and is exact for the $s$ states of the H atom. It compares favorably with the results given in the literature.