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.

在这个奖项下，PI将研究具有奇异度量结构的流形上的微分算子。其中一个项目是研究奇异空间上的波动方程。梅尔罗斯、维希和PI将研究波动方程的解通过复杂的奇异几何的绕射，也许最终会是一大类分层空间。这建立在梅尔罗斯之前的工作基础上，该工作表明，波动方程解的奇异性与锥点相互作用，产生从锥点发出的“绕射”球面波前。如果入射波前不是非常精确地聚焦在锥点上，则衍射波阵面的奇异性可能比入射波面的奇异性弱。另一个研究方向涉及流形上的薛定谔方程，其中无限的传播速度有一些有趣的结果。PI将研究囚禁几何中薛定谔演化的色散平滑效应的各个方面，最初专注于与拉格朗日子流形相关的正则性的传播。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有什么关系？对这个问题的洞察不仅来自对量子力学能量算符或“哈密顿”本身的直接研究，还来自与之相关的其他基本方程，如热方程、波动方程，当然还有含时间的薛定谔方程。因此，本研究的重点包括几类偏微分方程组的几何分析。一个项目研究了当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是非常微妙的。可能的物理应用包括“逆问题”，即人们试图通过观察穿过物体的波来推断物体的结构(例如地球内部)。另一个项目是研究弯曲空间上薛定谔方程的解的行为；这种解描述了量子粒子的时间演化。这一学科的进展也可能对非线性薛定谔方程产生影响，该方程出现在非线性光学和玻色-爱因斯坦凝聚理论等物理应用中。