Linear Partial Differential Equations on Singular Spaces

题目：奇异空间上的线性偏微分方程。i .: Jared Wunsch, Northwestern university . abstractpi将研究具有奇异度量结构的流形上的微分算子。一个项目关注奇异空间上的波动方程；与梅尔罗斯一起，PI已经表明，在一般情况下，波动方程解的奇点与锥点相互作用，产生从锥点发出的“衍射”球形波前。如果入射锋不太精确地聚焦于锥点，那么衍射锋的奇异性可能比入射锋弱。梅尔罗斯和PI希望将这些结果推广到更复杂的奇异几何上，也许最终包括一大类分层空间。另一个研究方向涉及具有“散射”度量的非紧流形上的薛定谔方程（如欧几里得空间的短程扰动）。与Hassell和Tao一起，PI参与了一个项目，以证明薛定谔方程在这种几何环境下的尖锐的strichartz估计。Hassell和PI也希望为薛定谔方程的传播子提供一个精确的描述，扩展一个产生部分Schwartz核的早期结果。量子力学数学理论中的一个中心问题是：粒子的经典动力学与其对应的量子态之间的关系是什么？对这个问题的见解不仅来自对量子力学能量算子“哈密顿算符”本身的直接研究，也来自对涉及它的许多其他基本方程的研究，比如热方程、波方程，当然还有时变薛定谔方程。因此，PI的研究重点是几种偏微分方程的几何分析。一个项目研究了当波与（某种泛化的）尖角相互作用时会发生什么——在这种情况下，由于衍射的影响，波前如何移动的几何形状可能相当微妙。另一个课题是研究无界空间上薛定谔方程解的结构；这样的解描述了量子粒子的时间演化。一个相关的研究重点是获得薛定谔方程解的某些估计，以提高我们对非线性薛定谔方程的理解，非线性薛定谔方程出现在非线性光学和玻色-爱因斯坦凝聚理论中，以及其他物理应用中。

Proposal DMS-0401323Title: Linear partial differential equations on singular spacesP.I.: Jared Wunsch, Northwestern UniversityABSTRACTThe PI will study differential operators on manifolds with singular metricstructures. One project focuses on the wave equation on singular spaces;together with Melrose, the PI has shown that in general, a singularity ofa solution to the wave equation interacts with a cone point to produce a"diffracted" spherical wavefront emanating from the cone point. Thesingularity of the diffracted front may be weaker than that of theincident front if the latter is not too precisely focused on the conepoint. Melrose and the PI hope to generalize these results to morecomplex singular geometries, perhaps eventually to include a large classof stratified spaces. Another direction of research involves theSchroedinger equation on noncompact manifold endowed with "scattering"metrics (e.g. short-range perturbations of Euclidean spaces). Togetherwith Hassell and Tao, the PI is involved in a project to prove sharpStrichartz estimates for the Schroedinger equation in this geometricsetting. Hassell and the PI also hope to provide a precise description ofthe propagator for the Schroedinger equation, extending an earlier resultwhich 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 andits corresponding quantum states? Insights into this problem have comenot only from the direct study of the quantum mechanical energy operatoror "Hamiltonian" itself, but also from studying a host of otherfundamental equations involving it, such as the heat equation, the waveequation, and, naturally, the time-dependent Schroedinger equation. Thefocus of the PI's research is thus the geometric analysis of several kindsof partial differential equations. One project investigates what happensto waves as they interact with (a certain generalization of) sharpcorners---the geometry of how wavefronts move can be quite subtle in thesecases owing to the effects of diffraction. Another project is to studythe structure of solutions of the Schroedinger equation on unboundedspaces; such solutions describe the time-evolution of a quantum particle. A related focus of research is to obtain certain estimates of solutions tothe Schroedinger equation in order to improve our understanding thenonlinear Schroedinger equation, which arises in nonlinear optics and thetheory of Bose-Einstein condensates, among other physical applications.