Linear Partial Differential Equations on Singular Spaces

Northwestern University, Evanston IL

Investigators

Abstract

This project will study partial differential equations on singular spaces, with an emphasis on spectral and scattering theory. The propagation of waves on smoothly varying spaces is well understood in many respects, but the interaction with singularities--which might range from boundaries, to corners, to structures "at infinity"--presents many open problems. One of the project's components is related to the understanding of wave propagation on Kerr spacetimes (i.e., near rotating black holes). The principal investigator will study the distribution of quasi-normal modes, the ways that the black hole may "ring" with damped oscillations. The initial goal of this project is to obtain a rigorous description of the exponential decay rate of high frequency modes. The project also includes work on problems of local energy decay on Riemannian manifolds, for which the geometry of the infinite ends turns out to have a profound effect on the low frequency phenomena that may dominate energy decay. Another aspect of wave propagation of interest to the principal investigator is the infinite-speed propagation occurring in solutions to the Schrodinger equation. In settings in which geometric rays are trapped in a bounded region, little is known about the regularity of solutions. The principal investigator is intent on studying the effects of these trapped rays, as well as the effects of geometric singularities such as cone points on the propagation. Geometry influences the behavior of solutions to wave equations in many interesting and subtle ways. Following Newton, we know that light behaves in many regimes as if made of tiny particles. On the other hand, we also know that light can turn corners ("diffract") and that it tends to disperse. The effect of changes in geometry to changes in propagation of waves (be they light or sound or water or gravity waves, or the wave-functions describing quantum particles) is the central focus of this project's research. In particular, the principal investigator's work on quasi-normal modes for Kerr spacetimes is closely related to problems of intense interest in the physics community, as these modes are part of the signature of gravitational waves. The principal investigator's study of the linear Schrodinger equation is related to applications not just to the physics of nonrelativistic quantum particles, but also to the nonlinear Schrodinger equation, which models such disparate phenomena as laser pulses and superconductivity.