Linear Partial Differential Equations on Singular Spaces

Northwestern University, Evanston IL

Investigators

Abstract

Proposal DMS-0401323 Title: Linear partial differential equations on singular spaces P.I.: Jared Wunsch, Northwestern University ABSTRACT The PI will study differential operators on manifolds with singular metric structures. One project focuses on the wave equation on singular spaces; together with Melrose, the PI has shown that in general, a singularity of a solution to 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. Melrose and the PI hope to generalize these results to more complex singular geometries, perhaps eventually to include a large class of stratified spaces. Another direction of research involves the Schroedinger equation on noncompact manifold endowed with "scattering" metrics (e.g. short-range perturbations of Euclidean spaces). Together with Hassell and Tao, the PI is involved in a project to prove sharp Strichartz estimates for the Schroedinger equation in this geometric setting. Hassell and the PI also hope to provide a precise description of the propagator for the Schroedinger equation, extending an earlier result which yielded part of its Schwartz kernel. 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 studying a host of other fundamental equations involving it, such as the heat equation, the wave equation, and, naturally, the time-dependent Schroedinger equation. The focus of the PI's research is thus the geometric analysis of several kinds of partial differential equations. One project investigates what happens to waves as 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. Another project is to study the structure of solutions of the Schroedinger equation on unbounded spaces; such solutions describe the time-evolution of a quantum particle. A related focus of research is to obtain certain estimates of solutions to the Schroedinger equation in order to improve our understanding the nonlinear Schroedinger equation, which arises in nonlinear optics and the theory of Bose-Einstein condensates, among other physical applications.