Mathematical Sciences: Analysis of Multi-Particle Systems

本数学物理研究项目涉及多粒子哈密顿量的特征值和特征函数，特别是与原子和分子系统有关的特征值和特征函数。一个主题是高电荷分子离子的不稳定性。另一个令人感兴趣的领域涉及与物理上显著能量差异的估计有关的问题，例如结合能和电离势。这类问题中的一些问题与本征函数的行为密切相关，特别是不同能级下本征函数的比较。因此，也提出了研究电子间距离等物理量在不同能量状态下期望值之间的关系。最后，提出了一种解决密度泛函理论中一个长期存在的问题的新方法。这里的一般思想是使用复杂的数学来模拟微观物理系统的通常违反直觉的行为。在这个层次上，我们不需要用数字来表示物理量，这是常识性的方法，而是用函数和函数空间上的算子来表示。这种表示的核心是一个叫做哈密顿算符的算子，它与能量有关。在哈密顿量和系统的时间演化之间有一个非常简单的抽象关系。然而，在实践中利用这种关系并不那么简单；困难的部分是计算特征值（可能的能级）和特征函数（对应的状态），无论哈密顿函数是什么。罗斯凯教授的项目的一个显著特点是它关注的是哈密顿量，哈密顿量来自于一个不是不受控制的大量粒子相互作用的系统。

This research project in mathematical physics is concerned with several problems about the eigenvalues and eigenfunctions of multi-particle Hamiltonians, particularly those associated with atomic and molecular systems. One topic is the instability of highly charged molecular ions. Another area of interest concerns problems related to the estimation of physically significant energy differences, such as binding energies and ionization potentials. Some of the problems in this category are closely related to the behavior of eigenfunctions, particularly comparisons of eigenfunctions for different energy levels. Therefore it is also proposed to investigate the relationship between expectation values in states of different energies for such quantities as the interelectronic distance. Finally, a new approach to solving a long-standing problem in density functional theory is presented. The general idea here is the use of sophisticated mathematics to model the often counterintuitive behavior of microscopic physical systems. At this level, one needs to represent physical quantities not by numbers, which would be the commonsense approach, but rather by functions and by operators on spaces of functions. At the heart of such a representation is an operator called the Hamiltonian, which has to do with energy. There is a quite simple abstract relationship between the Hamiltonian and the time evolution of the system. To exploit this relationship is not so simple in practice, however; the hard part is the calculation of eigenvalues (possible energy levels) and eigenfunctions (corresponding states) for whatever Hamiltonian one is working with. A distinctive feature of Professor Ruskai's project is its focus on Hamiltonians that come from systems in which a not uncontrollably large number of particles interact with one another.