The object of the proposed research is the geometric study of scattering theory. This encompasses such apparently distant areas as many-body scattering and the Laplacian on symmetric spaces. Powerful geometric techniques recently adapted from the study of hyperbolic equations have already played a major role in the detailed study of many-body scattering, which describes how quantum particles interact, e.g. in an accelerator. The proposed project will employ these microlocal techniques to explore new problems in many-body scattering, and to bring insights to analysis on symmetric spaces. These problems include the detailed study of the behavior of particles at threshold energies, which are energy levels where new configurations of particles become accessible. For many chemical phenomena Planck's constant, h, may be considered small, motivating the study of semiclassical many-body phenomena. A particular example is the spectral shift function, which has been the subject of an ongoing joint project of the PI with Xue Ping Wang. Another area of proposed research, joint with Gunther Uhlmann, is finite energy inverse many-body scattering, i.e. whether one can determine the interaction between particles from the scattering matrices, which are objects describing the outcome of scattering experiments. Perhaps surprisingly, there are very algebraic objects which are analogous to many-body problems, namely higher rank symmetric spaces. The structure of so-called flats in these spaces is very similar to the structure of the configuration space in many-body scattering, with the walls of the Weyl chambers playing the role of collision planes. In a joint project, Rafe Mazzeo and the PI plan to use the constructive techniques from many-body scattering to obtain the full asymptotic behavior of various analytic objects on these spaces, showing that many of the phenomena observed there have their counterparts in much greater generality.
Indeed, many people are familiar with the following two descriptions of the propagation of light. First, in geometric optics, light propagates in straight lines, reflecting from surfaces according to Snell's law. That is, the angles of incidence and of reflection are the same, as if light consisted of little billiard balls. Second, light can be described by the wave equation, its propagation thus being similar to that of water waves. There is a close relationship between these two viewpoints. Namely, for solutions of the wave equation, the propagation of sharp signals (or `singularities' of signals), which are the signals used to carry information, is precisely described by the simpler geometric optics picture. There is a similar correspondence between classical and quantum mechanics in many-particle interactions. Namely, for quantum particles the evolution of wave functions `at infinity' is almost described by the classical picture, as was proved previously by the PI. The only divergence between the classical and the quantum pictures is the emergence of bound states. For example, for a system consisting of two protons and two electrons, one of the protons and one of the electrons can form a hydrogen atom -- which may in turn break up if an electron with high energy hits it. In this proposal the PI will in part use these results, and related tools, for further investigation of many-body phenomena, such as examining the quantum-classical correspondence more precisely for small Planck's constant, and analyzing whether one can determine the interactions between particles from the result of a scattering experiment, and in part to extend such results to more geometric settings. These results are thus related to important questions in physics and chemistry.
