Spectral Asymptotics for Non-self-adjoint Semiclassical Operators

Pi：James V.Ralston(For Htrik)，UCLADMS-0304970ABSTRACT这个建议提出了半经典状态下非自伴微分算子谱理论中的问题。在几种情形下，作者寻求紧区域上自伴算子小扰动本征值的渐近行为的精确估计。在密切相关的项目中，他计划将最近开发的技术应用于寻找与凸形障碍物相关的散射极点的渐近性(计数函数)和薛定谔算子的势垒顶部共振。这项工作研究了波的传播，其中某种形式的耗散或传播到无穷远的可能性会产生随时间衰减到零的波。这些衰减模的衰减率和频率被编码成与这些问题相关的复本征值或共振序列。研究这些序列的行为可以更好地理解衰减率和系统基本结构之间的关系。这种信息在无损检测中根据物体的共振频率确定物体的内部结构具有潜在的应用价值。

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.