Linear Partial Differential Equations on Singular Spaces

本课题主要研究奇异空间上的偏微分方程组，重点是谱理论和散射理论。波在平稳变化的空间中的传播在许多方面都是众所周知的，但与奇点的相互作用--可能范围从边界到角点或圆锥点，再到大规模结构(在无穷远)--提出了许多悬而未决的问题。首席研究员将研究波在某些弯曲时空上传播的渐近行为，例如广义相对论中出现的弯曲时空。目标是了解在远离源的地方观察到的辐射模式的长期行为。首席研究人员最近的工作在时空方面取得了类似于Minkowski空间的紧凑化的结果，他将继续在更广泛的几何范围内追求这样的结果。该项目还将研究不同几何环境下波源附近的波的衰减。特别是，在存在拐角或圆锥点的情况下，波的绕射是波快速衰减的潜在障碍；首席研究员将继续他在这一领域正在进行的研究，特别是圆锥点产生的共振问题。此外，他还将继续研究与完全可积系统相关的拉普拉斯算子的本征函数和准模的行为，以努力了解本征函数在多大程度上可以集中在Arnold-Liouville环面的子环面上。在完全可积的情况下，这些技巧还应该给出线性薛定谔方程解的新估计。几何以许多有趣和微妙的方式影响波动方程的解的行为。在牛顿之后，我们知道光在许多情况下的行为就像是由微小的粒子组成的；另一方面，我们也知道光可以转弯(“衍射”)，而且它往往会消散。几何结构的变化对波(无论是光、声、水、重力波，或者描述量子粒子的波函数)传播的变化的影响是这个项目的中心焦点。特别是，首席研究员关于弯曲时空中波的衰减的工作与物理界感兴趣的问题密切相关，例如，它给出了一个非常简化的引力波模型。他在线性薛定谔方程上密切相关的工作不仅与非相对论量子粒子的物理学有关，还与非线性薛定谔方程有关，后者对激光脉冲和超导等截然不同的现象进行了建模。

This project focuses on the study of partial differential equations on singular spaces, with an emphasis on spectral and scattering theory. The propagation of waves on smoothly varying spaces is well understood in many respects, but the interaction with singularities--which might range from boundaries, to corners or cone points, to large-scale structures "at infinity"--presents many open problems. The principal investigator will study the asymptotic behavior of waves propagating on certain curved spacetimes such as arise in the theory of general relativity. The goal is to understand the long-time behavior of the radiation pattern observed far away from a source. Recent work of the principal investigator has yielded results in spacetimes with a compactification similar to that of Minkowski space, and he will continue to pursue such results in a wider range of geometries. The project will also study the decay of waves near their source in different geometric settings. In particular, in the presence of corners or cone points, diffraction of waves is a potential obstruction to the rapid decay of waves; the principal investigator will continue his ongoing investigations in this area, especially into the question of resonances generated by cone points. Additionally, he will continue to study the behavior of eigenfunctions and quasimodes of the Laplace operator associated to completely integrable systems, in an effort to understand to what degree an eigenfunction can concentrate on a sub-torus of an Arnold-Liouville torus. These techniques should also yield new estimates for solutions to the linear Schrodinger equation in completely integrable settings.Geometry influences the behavior of solutions to wave equations in many interesting and subtle ways. Following Newton, we know that light behaves in many regimes as if made of tiny particles; on the other hand, we also know that light can turn corners ("diffract") and that it tends to disperse. The effect of changes in geometry to changes in propagation of waves (be they light or sound or water or gravity waves, or the wave-functions describing quantum particles) is the central focus of this project. In particular, the principal investigator's work on decay of waves in curved spacetimes is closely related to problems of intense interest in the physics community, for instance as it gives a much simplified model of gravitational waves. His closely related work on the linear Schrodinger equation is related to applications not just to the physics of nonrelativistic quantum particles, but also to the nonlinear Schrodinger equation, which models such disparate phenomena as laser pulses and superconductivity.