This project studies the flow of energy, and the behavior of vibrational modes, for various mathematical models of physical systems. The principal investigator is introducing new mathematical tools that extend the understanding of energy flow phenomena to regimes that cannot be handled by standard analytical tools. Emphasis is placed on using methods that lend themselves to computational analysis. There is a long history of harmonic analysis tools finding application in signal and image processing, and the investigator's work involves developing similar tools for the study of wave propagation. One focus of the project is the reflection of waves from convex objects. The goal is to obtain a more precise understanding of how the wave disperses as it interacts with the boundary. Applications include control on the degree to which vibrational modes can concentrate near the boundary of a convex object. The project has another focus in the study of seismic waves, and more generally waves in elastic media. Seismic waves can involve both transverse and longitudinal displacements, and these components of the wave generally propagate at different speeds. A goal of the project is to estimate the order to which these distinct modes interact with each other as they propagate through highly heterogeneous media, such as the mixture of materials occurring within the earth. Practical implications include estimates on the error for computational models that treat the modes separately, and whether it is necessary to include their interaction in order to attain an assigned degree of accuracy. New methods will also be used in the study of decaying vibrational modes, known as resonant states. Example of systems with resonant states include microwave cavities, and quantum mechanical systems with potential barriers. Tools from harmonic analysis are used to study the existence of resonances, and to relate the number of resonances to properties of the system. All results of this project will be disseminated online, through open access websites.
This project involves the use of harmonic analysis techniques to advance our understanding of waves and eigenfunctions in nonhomogeneous media. A main goal of the project is to show that, in various settings where the traditional mathematical methods of geometric optics do not apply, the rate of dispersion of waves is nevertheless the same as would be predicted by geometric optics. An example of a setting studied is rough media, modeled by manifolds with twice-differentiable metrics. The energy of waves passing through such media can scatter, and only imprecise knowledge on energy flow is available. Nevertheless, the principal investigator's work shows that one can obtain sufficient control on energy flow in such media to establish important results, such as dispersive estimates that are of interest in the fields of nonlinear wave and Schrodinger equations. A related application is bounding the degree to which eigenfunctions in such media can concentrate. Another example of significant importance is seismic waves, which can propagate at distinct speeds, depending on the nature of the initial displacement. The principal investigator's research investigates the transfer of energy between various seismic modes that is induced by the singularities in the media through which they propagate. Scattering of waves from convex obstacles is another focus of the project. In this part of the proposed research the goal is to obtain precise rates of energy decay in small regions of the boundary. The results would show that energy cannot concentrate near the boundary to a degree higher than predicted by geometric optics, and would lead to new dispersive estimates, with consequent results for the study of nonlinear equations on domains with obstacles. The investigator is also applying harmonic analysis to the study of resonances for Schrodinger operators with bounded potentials. A sharp relation between higher Sobolev regularity of the potential, and series expansions for the regularized heat trace, is established, which is then used to prove the existence of resonances for such potentials.
