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.
