Mathematical Sciences: Spectral Asymptotics of Toeplitz andPseudodifferential Operators

数学科学：Toeplitz 和伪微分算子的谱渐进

• 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

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.

将继续开展关于赝微分算子谱渐进性和托普利茨算子相关问题的早期工作。 最近在形式意义上获得的与伪微分算子相关的展开已被证明在某些情况下是渐近正确的。 实际上，人们试图根据某种通用近似来获得有关算子函数迹的信息，该通用近似采用函数相对于度量的积分的形式。 这是 Weyl 和 Szego 在本世纪早期关于经典算子的开创性工作的基础。 目前的理论已经为具有平滑符号的算子产生了 Szego 展开式。 当前研究的目的是开发具有非平滑符号的运算符的扩展。 这些运算符不是数学产物；而是数学产物。它们出现在应用数学的几个领域。 解决这个问题的方法将基于具有非光滑幅度函数的积分的平稳相位方法。 我们还将致力于将 Szego 的 Toeplitz 行列式极限定理推广到 n 域。 托普利茨矩阵自然地作为伪微分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。 如果能够建立正确的类比，一维理论中的一个关键不等式将对更高维度的调和分析产生重要影响。 为此将开展工作。 我们还将开展额外的工作，以建立函数微积分来支持非自共轭托普利茨算子的研究，重点是学习乘数变化时谱的分布情况。