Mathematical Sciences: Spectral Asymptotics of Toeplitz andPseudodifferential Operators

早期谱渐近性工作的继续 伪微分算子及其相关问题 将进行Toeplitz算子。 与伪微分相关的最新展开式 在形式意义上获得的算子已经被证明， 在某些情况下，渐近正确。 实际上，一个是 试图获得关于一个函数的踪迹的信息， 算子的一些普遍近似， 函数关于a的积分的形式 measure. 这是魏尔开创性工作的基础 和塞戈在世纪早期对古典主义的研究 运营商 目前的理论产生了Szego展开， 具有平滑符号的运算符。 当前的对象 研究的目的是发展非光滑算子的展开式 符号. 这样的运算符不是数学工件;它们 在应用数学的几个领域中出现。 的途径 这个问题将基于一种固定相的方法， 非光滑振幅函数积分 还将努力寻求一种普遍适用于 Toeplitz上Szego的n-球面精化极限定理 决定因素 Toeplitz矩阵进入这些研究 自然地作为一维离散类似物， 伪微分算子，其中乘数为 不连续的 一维空间中的一个重要不等式 理论将对谐波分析有重要意义 在更高的维度，如果正确的模拟可以 确立了习 将为此开展工作。 还将开展更多工作， 支持非自伴研究的泛函演算 Toeplitz算子，重点放在学习如何 当乘子改变时，频谱分布。

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.