hal-00721928, version 1

Dynamical Ionization Bounds for Atoms

Enno Lenzmann ¹, Mathieu LEWIN ²

(2012-07-30)

• 1:  Mathematisches Institut
• 
  Universität Basel Switzerland
• 2:  Laboratoire d'Analyse, Géométrie et Modélisation (AGM)
• http://www.u-cergy.fr/rech/agm
  CNRS : UMR8088 – Université de Cergy Pontoise France

Bibliographic reference

• Type of document: Documents without publication reference (Preprint)
• Subject:

  Mathematics/Analysis of PDEs

  Physics/Mathematical Physics

  Mathematics/Mathematical Physics

• Title: Dynamical Ionization Bounds for Atoms
• 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 Schrödinger equation for atoms.
• Fulltext language: English
• Production date: 2012-07-30
• ANR Project:

  Project Id

  NoNAP

• European project:

  Cordis number	258023
  Acronyme	MNIQS
  Title	Mathematics and Numerics of Infinite Quantum Systems
  Funded by	ERC
  Start date	2010-10-01
  End date	2015-09-30
  Call identifier	ERC-2010-StG_20091028

• hal-00721928, version 1
• http://hal.archives-ouvertes.fr/hal-00721928
• oai:hal.archives-ouvertes.fr:hal-00721928
• From: Mathieu Lewin <>
• Submitted on: Tuesday, 31 July 2012 09:57:26
• Updated on: Tuesday, 31 July 2012 09:57:26