Linear Partial Differential Equations on Singular Spaces

Northwestern University, Evanston IL

Investigators

Abstract

Under this award the PI will study differential operators on manifolds with singular metric structures. One project is to study the wave equation on singular spaces. Melrose, Vasy, and the PI will investigate the diffraction of solutions of the wave equation by complex singular geometries, perhaps eventually a large class of stratified spaces. This builds on previous work with Melrose, which showed that a singularity of a solution of the wave equation interacts with a cone point to produce a "diffracted" spherical wavefront emanating from the cone point. The singularity of the diffracted front may be weaker than that of the incident front if the latter is not too precisely focused on the cone point. Another direction of research involves the Schroedinger equation on manifolds, where the infinite speed of propagation has some intriguing consequences. The PI will investigate aspects of the dispersive smoothing effect for Schroedinger evolution in trapping geometries, focusing initially on the propagation of regularity associated to Lagrangian submanifolds. A central question in the mathematical theory of quantum mechanics is: what is the relationship between the classical dynamics of a particle and its corresponding quantum states? Insights into this problem have come not only from the direct study of the quantum mechanical energy operator or "Hamiltonian" itself, but also from other fundamental equations involving it, such as the heat equation, the wave equation, and, naturally, the time-dependent Schroedinger equation. The focus of this research consequently includes the geometric analysis of several kinds of partial differential equations. One project investigates what happens to waves when they interact with (a certain generalization of) sharp corners---the geometry of how wavefronts move can be quite subtle in these cases owing to the effects of diffraction. Possible physical applications include "inverse problems" in which one attempts to deduce the structure of an object (e.g. the interior of the earth) from observation of waves that have passed through it. Another project is to study the behavior of solutions of the Schroedinger equation on curved spaces; such solutions describe the time-evolution of a quantum particle. Progress in this subject may also have consequences for the nonlinear Schroedinger equation, which arises in nonlinear optics and the theory of Bose-Einstein condensates, among other physical applications.