Vol. 6, No. 5, 2013

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Dynamical ionization bounds for atoms

Enno Lenzmann and Mathieu Lewin

Vol. 6 (2013), No. 5, 1183–1211

We study the long-time behavior of the 3-dimensional repulsive nonlinear Hartree equation with an external attractive Coulomb potential Z|x|, 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 4Z (2Z 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 |x|3) 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.

Hartree equation, RAGE theorem, ionization bound, positive commutator
Mathematical Subject Classification 2010
Primary: 35Q55, 81Q05, 81Q10, 35Q41
Received: 20 November 2012
Accepted: 29 April 2013
Published: 3 November 2013
Enno Lenzmann
Mathematisches Institut
Universität Basel
Rheinsprung 21
CH-4051 Basel
Mathieu Lewin
Laboratoire de Mathématiques
Université de Cergy-Pontoise
95000 Cergy-Pontoise