Professor Soffer's project is concerned with the spectral and scattering theory of hamiltonian systems. It includes as a major part the study of N-body long range potential systems and quasiparticle dynamics such as spin waves. Related problems will also be tackled, including the spectral theory of time-dependent hamiltonians and multichannel nonlinear scattering. It is expected that this work will provide a rigorous basis for some important concepts in physics as well as new mathematical techniques of analyzing the asymptotic behavior of solutions of partial differential equations. The general idea here is the use of sophisticated mathematics to model the behavior of physical systems, especially those involving several interacting particles. (These might be electrons interacting by Coulomb forces, or at the opposite extreme, stars interacting gravitationally.) Such a system may variously be described by a system of partial differential equations, or by an operator (called the hamiltonian) on a space of functions that represent the possible physical states. There is in principle a quite straightforward relationship between the hamiltonian and the time evolution of the system. Working this out for a given class of hamiltonians in sufficient detail to make strong qualitative statements about what happens to the system in the long run is a major analytical task, however, and the focus of the mathematical research for this project.
