Mathematical Sciences: Scattering Theory for Linear and Non-linear Equations

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This research concerns scattering theory and, in particular, the study of scattering poles or resonances and nonlinear diffraction. The resonances which in a wide range of situations appear as the poles of the meromorphic continuation of the resolvent of the operator constitute, in some sense, are the replacement of the discrete spectral data for problems on noncompact domains. As in the case of eigenvalues, the link between the distribution of these poles and the dynamical properties of the scatterer is of great interest. Also, this project supports the study of the regularity of nonlinear waves on domains with diffractive boundaries which represents a natural continuation of the well understood linear theory. The main concern is an accurate description of the location of the 'anomalous' singularities due to nonlinear interaction. Finally, this project will involve a study of operators arising in solid state physics. This topic is concerned with the spectral theory rather than the scattering theory but it shares the microlocal aspects of the scattering theory. This research, in the general area of geometry, involves the behavior of waves and wave fronts together with the spectral problems motivated by solid-state physics and noncommutative geometry.