Dynamical ionization bounds for atoms
Dynamical ionization bounds for atoms
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原子的动态电离界限
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Mathieu Lewin
中科院分区：
文献类型：
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作者：
E. Lenzmann;Mathieu Lewin
研究了具有外部吸引库仑势$-Z/的三维排斥非线性Hartree方程的长时间行为|X| $，这是一个原子量子动力学的非线性模型。我们表明，经过足够长的时间，在任何有限球的平均电子数总是小于4 Z（分别为2 Z的径向情况下）。这是E.H. Lieb关于静止情况下原子的最大负电离。我们的证明涉及到一个新的积极交换子论点（基于立方重量$|X| ^3 $），我们的发现让人想起了爱因斯坦定理。此外，我们证明了一个类似的普遍约束的局部动能。特别是，我们的主要结果意味着，在弱意义上，任何解决方案被吸引到一个有界的能量空间，无论初始数据的大小。此外，我们还将主要结果推广到Hartree-Fock理论和原子的线性多体薛定谔方程。
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 Schrodinger equation for atoms.