Mathematical Sciences: Spectral Asymptotics of Toeplitz andPseudodifferential Operators

关于拟微分算子的谱的渐近性和有关Toeplitz算子的相关问题的工作将继续进行。最近得到的与形式意义上的拟微分算子有关的展开式在某些情况下已被证明是渐近正确的。实际上，人们试图以某种通用近似的形式获得关于算子的函数的迹的信息，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在本世纪初关于经典算子的开创性工作的基础。目前的理论已经给出了具有光滑符号的算子的Szego展开。本研究的目的是建立具有非光滑符号的算子的展开。这样的运算符不是数学产物；它们出现在应用数学的几个领域。这个问题的解决方法将基于具有非光滑振幅函数的积分的驻相方法。在寻求Szego关于Toeplitz行列式的精炼极限定理的n-球面的推广方面，也将进行工作。Toeplitz矩阵自然而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的不等式将对高维的调和分析产生重要的影响。我们将为此开展工作。为了建立函数演算来支持对非自伴Toeplitz算子的研究，还将进行更多的工作，重点是了解当乘子改变时光谱如何分布。

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.