Mathematical Sciences: Spectral Asymptotics of Toeplitz andPseudodifferential Operators

数学科学：Toeplitz 和伪微分算子的谱渐进

基本信息

批准号：
8822906
负责人：
Harold Widom
金额：
$ 8.14万
依托单位：
University of California-Santa Cruz
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1989
资助国家：
美国
起止时间：
1989-07-01 至 1993-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=8822906&HistoricalAwards=false
关键词：
Mathematical Sciences Spectral Asymptotics Toeplitz

项目摘要

A continuation of earlier work on the spectral asymptotics of pseudodifferential operators and related questions on Toeplitz operators will be carried out. Recent expansions associated with pseudodifferential operators obtained in a formal sense have been shown, in certain cases, to be correct asymptotically. In effect, one is trying to gain information about the trace of functions of an operator in terms of some universal approximation which takes the form of an integral of the function with respect to a measure. This was the basis for the pioneering work of Weyl and Szego in the earlier part of the century on classical operators. Present theory has produced a Szego expansion for operators with a smooth symbol. The object of the current study is to develop expansions for operators with nonsmooth symbols. Such operators are not mathematical artifacts; they arise in several areas of applied mathematics. The approach to this question will be based on a method of stationary phase for integrals with nonsmooth amplitude function. Work will also be done in seeking a generalization to the n-sphere of Szego's refined limit theorem on Toeplitz determinants. Toeplitz matrices enter into these studies naturally as one-dimensional discrete analogues of pseudodifferential operators, where the multiplier is discontinuous. A crucial inequality in the one-dimensional theory would have important implications to harmonic analysis in higher dimensions if the correct analogue can be established. Work will be done towards this end. Additional work will be done in an effort to establish a functional calculus to support the study of nonselfadjoint Toeplitz operators, with emphasis placed on learning how the spectra distribute when the multipliers change.

将继续对伪差异操作员的光谱渐近学工作以及有关Toeplitz操作员的相关问题的延续。在某些情况下，与正式意义上获得的伪差异操作员相关的最新扩展是渐近地是正确的。实际上，人们正在尝试以某些通用近似为操作员的功能轨迹获取信息，该通用近似以量度相对于度量的函数积分的形式。这是本世纪初期关于古典运营商的开创性工作的基础。当前的理论为具有光滑符号的操作员提供了Szego扩展。当前研究的目的是为具有非平滑符号的操作员开发扩展。这样的操作员不是数学工件。它们出现在应用数学的几个领域。该问题的方法将基于具有非平滑幅度函数积分的固定相的方法。还将完成对Szego关于Toeplitz决定因素的精制限制定理的N-Sphere的概括。 Toeplitz矩阵自然地作为伪差算子的一维离散类似物进入这些研究，其中乘数是不连续的。如果可以建立正确的类似物，则一维理论中的至关重要不平等将对更高维度的谐波分析具有重要意义。工作将在这一目标上完成。将进行其他工作，以建立一个功能性计算，以支持对非夫拉多的toeplitz运算符的研究，重点是学习乘数何时变化时光谱如何分发。