Linear Partial Differential Equations on Singular Spaces

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Many seemingly different physical phenomena such as light, sound propagation, and the motion of quantum particles are described in a mathematically unified manner as propagating waves. It is therefore of considerable practical importance to describe how waves oscillate and die off depending on the source producing them and the medium in which they propagate. The goal of this project is to investigate several questions surrounding the rate of wave decay, and in particular how it is influenced by the effects of diffraction, which occurs when the waves encounter sharp discontinuities in the medium. The questions of interest include those motivated by very small-scale physics (the hydrogen atom) and very large-scale physics (decay of waves on backgrounds arising in cosmology). The PI’s prior work on analyzing the effectiveness of computational methods used in modeling solutions to these equations in practical settings will be continued. Training of graduate students and postdoctoral fellows will be incorporated throughout the project.This project revolves around questions involving the decay rate of waves near their source in several settings. Of particular interest will be understanding the role of wave diffraction by rough media in the qualitative behavior and long-time decay rates of solutions to wave and Schrödinger equations. When studying the effects of the singularity of the Coulomb potential on the structure of the Dirac propagator for the hydrogen atom, diffractive effects will again play an important role. In spacetimes of interest in general relativity, the large-scale structure of spacetime effects on the decay of waves and the structure of their radiation patterns will be investigated. This project will further entail a study of the performance of numerical algorithms for computation of the scattering of waves, bringing to bear techniques of phase space analysis that have not previously been employed in these problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.