Mathematical Sciences: Analysis of Multi-Particle Systems

点击翻译按钮获取中文摘要

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.