Linear Partial Differential Equations on Singular Spaces

The problem of understanding wave propagation is ubiquitous in mathematics and its physical applications: sound waves, light waves, ripples in the structure of spacetime, and the "wave-functions" that describe particles on a microscopic scale are all described by a family of closely related equations. When the media in which the waves propagate are rough or discontinuous, wave propagation is complicated by the effects of diffraction. This project is concerned with a number of aspects of wave propagation in settings where diffraction may play a novel role. The Principal Investigator (PI) will address fundamental questions such as: how do sound waves resonate in a medium where the sound speed may jump? how is a quantum particle affected by discontinuities in the underlying medium? how are internal waves in the ocean scattered by variation in the structure of the bottom? Potential areas of application include better design of quantum devices, improved control and damping of acoustic waves, and better understanding of the mixing of the ocean. The project provides research training opportunities for graduate students and postdoctoral scholars.

The PI investigates questions involving wave equations in several settings with the goal of understanding the behavior of the wavefunctions arising in quantum mechanics when the potential function that governs the system may not vary smoothly. These results will also apply to the behavior of acoustic waves in settings where the sound-speed may not be smooth. Further, the PI studies the scattering of internal waves in the ocean as they propagate over a variable bottom topography. The effects on quantum scattering theory of diffraction from multiple Coulomb potentials are also analyzed.

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.

Award Number: 2452331
Principal Investigator: Jared Wunsch
Funds Obligated: $200,000
State: IL