Spectral Asymptotics for Non-self-adjoint Semiclassical Operators

主要研究者：Ralston（for Hatrik），UCLADMS-0304970摘要该方案提出了半经典区域中非自伴微分算子谱理论的问题。提出者寻求精确的估计的小扰动的自伴算子的特征值的渐近行为的紧域上在几个设置。在密切相关的项目中，他计划应用最近开发的技术来寻找问题的渐近性（计数功能）的散射极点与凸障碍和障碍顶部共振的薛定谔operators.This工作研究波的传播设置中的某种形式的耗散或传播到无穷大的可能性产生的波衰减到零随着时间的增加。这些衰减模式的衰减率和频率被编码在与这些问题相关联的复特征值或共振序列中。研究这些序列的行为可以更好地理解衰变率和系统的基本结构之间的关系。这些信息在无损检测中从物体的共振频率确定物体的内部结构方面具有潜在的应用。

PI: James V. Ralston (for Hatrik), UCLADMS-0304970ABSTRACT This proposal presents problems in spectral theory of non-self-adjoint differential operators in the semi-classical regime. The proposer seeks precise estimates on the asymptotic behavior of the eigenvalues of small perturbations of self-adjoint operators on compact domains in several settings. In closely related projects he plans to apply recently developed techniques to the problem of finding asymptotics (counting functions) for the scattering poles associated withconvex obstacles and the barrier top resonances for Schroedinger operators.This work studies the propagation of waves in settings where some form of dissipation or the possibility of propagation to infinity gives rise to waves which decay to zero as time increases. The rates of decay and frequency of these decaying modes are encoded in sequences of complex eigenvalues or resonances associated with these problems. Studying the behavior of these sequences can lead to better understanding of the relation between rates of decay and theunderlying structure of the system. Such information has potential application in determining the interior structure of objects from their resonant frequencies in nondestructive testing.