Mathematical Sciences: Scattering Theory for N-particle Systems

DMS-9501033 PI：N粒子体系的拉尔斯顿散射理论本研究的主题是具有恒定磁场的N粒子薛定谔算子的散射理论。与不存在磁场的情况(现在已经很好地理解了)相反，尽管它具有数学和物理意义，但人们以前对它知之甚少。主要任务将是证明这类算子的渐近完备性，即根据相应的含时薛定谔方程的大时间渐近行为，给出其所有解的完全分类。最近，C.Gerard和作者证明了不含真中立子系统(包括原子和正离子)的N体系统的渐近完备性。这项拟议研究的目的是将这些结果推广到一般N-体系统的情况，包括负离子和分子。这里的主要问题将是分析中立子系统的色散行为。它们的动力学依赖于它们的内部结构，因此在扰动下可能非常不稳定。还需要对与磁场有关的几何结构有更深入的了解。这项研究涉及磁场的存在如何影响结构物质的基本问题。关于这个问题的几乎所有早期结果都是在简化的假设下得到的，即原子核的运动可以忽略不计；从数学上讲，这意味着研究原子核具有无限质量的近似。然而，这个简化的模型在物理学上并不总是合适的，特别是因为我们的研究表明，这个模型的性质与一个更现实的有限核质量模型之间存在着重要的定性差异。此外，尽管这个问题在物理学和天体物理学中引起了相当大的兴趣(例如，预计中子星表面存在强磁场)，但无法根据实验做出预测。这样做的原因是，在实验室中可以产生的磁场不够强，无法检测到它们对有关系统的影响。因此，人们不得不依赖于对问题的理论分析。

DMS-9501033 PI: Ralston Scattering theory for N-particle systems The subject of the proposed research is scattering theory for N-particle Schrodinger operators with constant magnetic fields. In contrast to the case when no magnetic field is present (which is now well understood), very little was previously known about it despite its mathematical and physical significance. The main task will be to prove asymptotic completeness for such operators, i.e., to give a complete classification of all solutions of the corresponding time-dependent Schrodinger equation according to their large-time asymptotic behaviour. Recently C. Gerard and the author proved asymptotic completeness for N-body systems containing no proper neutral subsystems (this class includes atoms and positive ions). The aim of the proposed research is to extend these results to the case of general N-body systems, including negative ions and molecules. The main problem here will be to analyze the dispersive behaviour of the neutral subsystems. Their dynamics depends on their internal structure and therefore may be very unstable under perturbations. A deeper understanding of the geometrical structures related to the magnetic field will also be required. This research deals with fundamental questions about how the structure matter is affected by the presence of a magnetic field. Almost all of the earlier results on this subject were obtained under the simplifying assumption that the motion of the nuclei is negligible; mathematically, this means studying an approximation where nuclei have infinite mass. However, this simplified model is not always appropriate in physics, especially since our research has shown that there are important qualitative differences between the properties of this model and a more realistic one with finite nuclear masses. Furthermore, although this problem is of considerable interest in physics and astrophysics (for example, strong magnetic fields are ex pected to exist on the surfaces of neutron stars), no predictions could be made on the basis of experiments. The reason for this is that the magnetic fields that can be generated in the labolatories are not strong enough for their effect on the systems in question to be detected. One thus has to rely on a theoretical analysis of the problem.