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2013 Dynamical ionization bounds for atoms
Enno Lenzmann, Mathieu Lewin
Anal. PDE 6(5): 1183-1211 (2013). DOI: 10.2140/apde.2013.6.1183


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.


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Enno Lenzmann. Mathieu Lewin. "Dynamical ionization bounds for atoms." Anal. PDE 6 (5) 1183 - 1211, 2013.


Received: 20 November 2012; Accepted: 29 April 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1284.35404
MathSciNet: MR3125553
Digital Object Identifier: 10.2140/apde.2013.6.1183

Primary: 35Q41, 35Q55, 81Q05, 81Q10

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.6 • No. 5 • 2013
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