Many-Body Scattering

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The object of the proposed research by Andras Vasy is the study of many-bodyscattering. The data from a scattering experiment are mathematicallyrepresented by the scattering matrices, or S-matrices. The central issuein many-body scattering is understanding the structure of these operators.A new method is now available for the analysis; it adopts techniques usedfor the study of hyperbolic equations. This project will employ thesemicrolocal techniques to analyze the wave front relation of the free-clusterto free-cluster scattering matrix in many-body scattering, involvingarbitrarily many particles, when the subsystems have no bound states.This S-matrix represents scattering data when both the incoming and theoutgoing particles are asymptotically free, i.e. they are not bound together.Its wave front relation describes the singularities of the outgoing datain terms of those of the incoming ones. More precisely, the main goal of thepresent project is to prove that if the subsystems of a many-body systemhave no bound states, e.g. if all potentials are non-negative, then the wavefront relation of the free-cluster to free-cluster S-matrix is given bythe broken geodesic flow to distance pi on the sphere. The absence of boundstates simplifies the phase space structure of the problem and henceprovides an ideal first step towards the understanding of the S-matricesin general. In addition, in this situation the problem is geometricallyanalogous to the wave equation in domains with corners. Thus, the completionof the project will demonstrate that scattering is in many respects ahyperbolic problem, similar to the wave equation. The next step will beto analyze what happens if there are bound states in some subsystems;in this case one has to combine the behavior of bound states with theclassical dynamics to understand the propagation of singularities.Indeed, many people are familiar with the following two descriptions ofthe propagation of light. First, in geometric optics, light propagatesin straight lines, reflecting from surfaces according to Snell's law.That is, the angles of incidence and of reflection are the same, as iflight consisted of little billiard balls. Second, light can be described bythe wave equation, its propagation thus being similar to that of waterwaves. There are many ways in this particular example in which the firstpicture gives a rough description of the second, but the followingconnection has particularly wide-ranging generalizations: singularities ofthe amplitude of the wave (`sharp signals' or `jumps in signals') propagatealong light rays. The purpose of the current proposal is to investigate theanalogous relationship between classical and quantum mechanical particles,that is, to analyze in what ways does the motion of classical particles,considered as little billiard balls colliding with each other, describethe motion of quantum mechanical particles. In more concrete terms, how welldoes the motion of four billiard balls describe how four electrons collidewith each other? Since it is easier to analyze the corresponding propertiesof classical systems than those of quantum ones, just as it is easier todescribe light moving in straight lines than to solve the wave equation,the answer to this question significantly improves our understanding ofquantum mechanics.