Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 0700318
• 负责人: Jared Wunsch
• 金额: $ 12.7万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700318&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

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.

根据该奖项，PI将研究具有奇异度量结构的流形的差异操作员。一个项目是研究奇异空间上的波动方程。 Melrose，Vasy和Pi将通过复杂的奇异几何形状研究波动方程的溶液的衍射，也许最终可能是一大批分层空间。这是基于先前与梅尔罗斯（Melrose）的作品建立的，这表明波动方程的溶液的奇异性与锥体点相互作用，从而产生了从锥点发出的“衍射”球形波前。如果不太精确地专注于锥体点，则衍射前部的奇异性可能比入射正面的奇异性要弱。研究的另一个方向涉及流形方程，其中无限的传播速度具有一些有趣的后果。 PI将研究Schroedinger进化的分散平滑效应在捕获几何形状中的各个方面，最初集中于与Lagrangian Submanifold相关的规则性的传播。对这个问题的见解不仅来自对量子机械能运算符或“哈密顿量”本身的直接研究，而且还来自涉及它的其他基本方程，例如热方程，波动方程，以及自然而然地依赖时间依赖时间的Schroedinger方程。因此，这项研究的重点包括对几种偏微分方程的几何分析。一个项目调查了波浪与尖角（某个概括）相互作用时发生的事情 - 在这些情况下，由于衍射的影响，波前的运动方式的几何形状可能非常微妙。 可能的物理应用包括“反问题”，其中试图通过观察到通过它的波的观察来推断物体的结构（例如地球内部）。 另一个项目是研究Schroedinger方程在弯曲空间上的解决方案的行为。这样的解决方案描述了量子粒子的时间进化。 该主题的进展也可能对非线性Schroedinger方程式产生后果，该方程在非线性光学和Bose-Einstein冷凝水理论中产生，以及其他物理应用。