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

本研究涉及散射理论，特别是散射极点或共振和非线性衍射的研究。在广泛的情况下，共振作为算子构成的解的亚纯延拓的极点，在某种意义上是对非紧域上问题的离散谱数据的替代。在本征值的情况下，这些极点的分布和散射体的动力学性质之间的联系是非常有趣的。此外，该项目还支持对具有衍射边界的域上非线性波的规律性的研究，这代表了众所周知的线性理论的自然延续。主要关注的是由于非线性相互作用导致的“异常”奇点位置的准确描述。最后，本计画将涉及固态物理中出现的算子的研究。本课题关注的是光谱理论而不是散射理论，但它分享了散射理论的微局部方面。本研究在一般的几何领域，涉及波和波前的行为，以及由固态物理和非交换几何引起的光谱问题。

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.