Linear Partial Differential Equations on Singular Spaces

Overview The PI proposes to study partial differential equations on singular spaces, with an emphasis on spectral and scattering theory. The propagation of waves in smoothly varying media is generally well understood, but the interaction of waves with singularities, for instance the effects of diffraction by discontinuous media, rough boundaries, or large electromagnetic potentials, remains a very active area of research. The PI hopes to obtain new results which will enhance understanding of the long-time behavior of waves as well as the numerical methods used in simulating them. Intellectual Merit: The PI will investigate questions involving the rate of decay of waves near their source in several settings. In particular, he will work to understand the role that diffraction of waves by rough media plays in the qualitative behavior and long-time decay rates of solutions to the wave and Schrodinger equations. He will study the effects of the singularity of the Coulomb potential on the structure of the Dirac propagator for the hydrogen atom, where diffractive effects again play an important role. In spacetimes of interest in General Relativity, he will investigate how the large-scale structure of spacetime affects the decay of waves and the structure of their radiation patterns. The PI will also study 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. Broader Impacts: The PI’s work on decay of waves has potential applications in inverse problems relevant to seismology and medical imaging. Some of this work is also directly motivated by problems from numerical analysis of high-frequency scattering problems, and will improve understanding of how best to simulate electromagnetic, acoustic, and other types of wave propagation. The PI will integrate the training of students and postdocs into the proposed research. He will continue to be active in organizing seminars, conferences, and summer schools in the field.