Mathematical Sciences: Scattering Theory for N-particle Systems

DMS-9501033 PI：N粒子系统的Ralston散射理论 本文的研究对象是N粒子的散射理论 具有恒定磁场的薛定谔算子。相对于 在没有磁场存在的情况下（现在已经很好地理解了）， 以前对它知之甚少，尽管它的数学和 物理意义。主要任务是证明渐近性 这种算子的完备性，即，进行完整的分类 相应的含时薛定谔方程的所有解 根据它们的大时间渐近行为。最近C。杰拉德和 作者证明了不含N体系统的渐近完备性， 适当的中性子系统（这类包括原子和正离子）。 所提出的研究的目的是将这些结果扩展到 一般的N体系统，包括负离子和分子。主要 这里的问题将是分析中性的色散行为 子系统它们的动力学取决于它们的内部结构， 因此在扰动下可能非常不稳定。了更深刻的认识 与磁场有关的几何结构也将被 必需的. 这项研究涉及的基本问题是， 物质受到磁场的影响。几乎所有 关于这一问题的早期结果是在简化条件下得到的。 假设原子核的运动可以忽略不计;数学上， 这意味着研究原子核具有无限质量的近似。 然而，这种简化的模型在物理学上并不总是合适的， 特别是我们的研究表明， 该模型的属性之间的质的差异， 一个具有有限核质量的现实模型。此外，虽然这 这个问题在物理学和天体物理学中具有相当大的兴趣（因为 例如，强磁场被认为存在于表面上 中子星），没有预测可以在实验的基础上。这是因为磁场可以 在实验室产生的是不够强大，他们的影响， 有问题的系统被检测。因此，人们必须依靠 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.