Linear Partial Differential Equations on Singular Spaces

建议DMS-0401323标题：奇异空间上的线性偏微分方程组P.I.：Jared Wunsch，西北大学摘要PI将研究具有奇异度量结构的流形上的微分算子。一个项目集中于奇异空间上的波动方程；PI与Melrose一起表明，一般而言，波动方程解的奇异性与圆锥点相互作用，产生从圆锥点发出的“绕射”球面波前。如果入射波阵面不是太精确地聚焦在点上，那么衍射波阵面的单一性可能比入射波阵面的单一性弱。梅尔罗斯和PI希望将这些结果推广到更复杂的奇异几何，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(例如欧氏空间的短程扰动)。与哈塞尔和陶一起，PI参与了一个项目，以证明薛定谔方程在这种几何背景下的精确的Strichartz估计。Hassell和PI还希望为薛定谔方程的传播子提供精确的描述，推广了早期得出其部分Schwartz核的结果。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相应的量子态之间是什么关系？对这个问题的见解不仅来自于对量子力学能量算符或“哈密顿”本身的直接研究，还来自于对与之相关的许多其他基本方程的研究，如热方程、波动方程，当然还有含时薛定谔方程。因此，PI的研究重点是几类偏微分方程的几何分析。一个项目研究了波与尖角相互作用时会发生什么-由于绕射的影响，波前如何移动的几何在这些情况下可能非常微妙。另一个项目是研究无界空间上薛定谔方程的解的结构；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，该方程产生于非线性光学和玻色-爱因斯坦凝聚理论等物理应用中。

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.