We study the long-time behavior of the 3-dimensional repulsive nonlinear
Hartree equation with an external attractive Coulomb potential
, which is
a nonlinear model for the quantum dynamics of an atom. We show that, after a
sufficiently long time, the average number of electrons in any finite ball is always smaller
than
( in
the radial case). This is a time-dependent generalization of a celebrated result by
E.H. Lieb on the maximum negative ionization of atoms in the stationary case. Our
proof involves a novel positive commutator argument (based on the cubic weight
) and
our findings are reminiscent of the RAGE theorem.
In addition, we prove a similar universal bound on the local kinetic energy. In
particular, our main result means that, in a weak sense, any solution is attracted to a
bounded set in the energy space, whatever the size of the initial datum. Moreover, we
extend our main result to Hartree–Fock theory and to the linear many-body
Schrödinger equation for atoms.