Scattering Theory on Manifolds

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The P.I. studies problems in scattering theory. Some involve resonances, complex numbers which for many problems on noncompact domains are natural analogs of eigenvalues. Of particular interest to the P.I. are lower bounds on the number of resonances and the existence of asymptotic expansions of solutions to the wave equation in terms of resonances when there is trapping. The investigator will also study the problem of determining the asymptotic expansion of a perturbation of a stratified sound speed from knowledge of the scattering matrix at fixed energy. Another problem of interest is understanding the relationship between the sojourn times of geodesics and the scattering matrix for manifolds with cylindrical ends. The investigator studies questions that are related to wave propagation. Resonances can be viewed as giving the energy and rate of decay of waves. The P.I. studies the distribution of resonances and their relationship to the long-time behaviour of waves. Another area of interest to the P.I. is inverse scattering theory, which involves recovering information about a an obstacle or perturbation from measurements made "far away" of waves affected by the obstacle. Taken in a broad sense, this has found many applications, including radar, medical imaging, and underground exploration.