Dynamical Ionization Bounds for Atoms

TL;DR

This paper establishes universal bounds on the long-time electron distribution and kinetic energy for atoms modeled by nonlinear and linear quantum equations, extending classical results to dynamic settings.

Contribution

It introduces a novel positive commutator method to prove time-dependent ionization bounds for atoms in various quantum models, including Hartree, Hartree-Fock, and Schrödinger equations.

Findings

Average electrons in any finite region are bounded by 4Z (or 2Z radially) over time.

Universal bounds on local kinetic energy are established.

Solutions tend to remain within a bounded energy set over time.

Abstract

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 (respectively 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 Schr\"odinger equation for atoms.