Abstract
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.
Citation
Enno Lenzmann. Mathieu Lewin. "Dynamical ionization bounds for atoms." Anal. PDE 6 (5) 1183 - 1211, 2013. https://doi.org/10.2140/apde.2013.6.1183
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