Linear Partial Differential Equations on Singular Spaces

本专题将研究奇异空间上的偏微分方程，重点是光谱和散射理论。波在平滑变化的空间中的传播在许多方面都得到了很好的理解，但是与奇点的相互作用-可能从边界到角落，到“无限远”的结构-提出了许多开放的问题。该项目的一个组成部分是关于克尔时空上波传播的理解（即，旋转黑洞附近）。首席研究员将研究准正常模式的分布，即黑洞可能以阻尼振荡“振铃”的方式。这个项目的最初目标是获得一个严格的描述指数衰减率的高频模式。该项目还包括对黎曼流形上的局部能量衰减问题的研究，其中无限端的几何形状对可能主导能量衰减的低频现象产生了深远的影响。波传播的另一个方面的主要研究者感兴趣的是无限速度传播发生在解决方案的薛定谔方程。在几何光线被困在一个有界区域的设置，很少有人知道的规则性的解决方案。首席研究员致力于研究这些被捕获的射线的影响，以及几何奇点（如锥点）对传播的影响。几何以许多有趣而微妙的方式影响波动方程解的行为。遵循牛顿的理论，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（“转弯”），而且它倾向于分散。几何变化对波传播变化的影响（无论是光波、声波、水波、重力波，还是描述量子粒子的波函数）是该项目研究的中心焦点。特别是，首席研究员对克尔时空的准正常模式的研究与物理界的强烈兴趣问题密切相关，因为这些模式是引力波签名的一部分。主要研究者对线性薛定谔方程的研究不仅与非相对论量子粒子的物理学有关，而且与非线性薛定谔方程有关，后者模拟了激光脉冲和超导等不同的现象。

This project will study 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, to structures "at infinity"--presents many open problems. One of the project's components is related to the understanding of wave propagation on Kerr spacetimes (i.e., near rotating black holes). The principal investigator will study the distribution of quasi-normal modes, the ways that the black hole may "ring" with damped oscillations. The initial goal of this project is to obtain a rigorous description of the exponential decay rate of high frequency modes. The project also includes work on problems of local energy decay on Riemannian manifolds, for which the geometry of the infinite ends turns out to have a profound effect on the low frequency phenomena that may dominate energy decay. Another aspect of wave propagation of interest to the principal investigator is the infinite-speed propagation occurring in solutions to the Schrodinger equation. In settings in which geometric rays are trapped in a bounded region, little is known about the regularity of solutions. The principal investigator is intent on studying the effects of these trapped rays, as well as the effects of geometric singularities such as cone points on the propagation.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's research. In particular, the principal investigator's work on quasi-normal modes for Kerr spacetimes is closely related to problems of intense interest in the physics community, as these modes are part of the signature of gravitational waves. The principal investigator's study of 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.