Mathematical Sciences: Spectral Asymptotics of Toeplitz andPseudodifferential Operators

伪微分算子的谱渐近性和关于Toeplitz算子的相关问题的早期工作的延续将进行。最近在形式意义上得到的与伪微分算子相关的展开式，在某些情况下是渐近正确的。实际上，人们试图获得关于一个算子的函数轨迹的信息，以某种普遍近似的形式，这种近似的形式是函数对一个测度的积分。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。目前的理论给出了具有光滑符号的算子的Szego展开。本研究的目的是发展具有非光滑符号的算子的展开式。这样的运算符不是数学上的工件；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非光滑振幅函数积分的定相方法。我们还将研究如何将Szego关于Toeplitz行列式的精炼极限定理推广到n球。Toeplitz矩阵作为伪微分算子的一维离散类似物自然地进入这些研究，其中乘子是不连续的。如果能建立正确的类比，一维理论中的一个关键不等式将对高维谐波分析产生重要影响。我们将为此而努力。额外的工作将是努力建立一个函数演算来支持非自伴随Toeplitz算子的研究，重点是学习当乘子变化时谱的分布。

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.