Mathematical Sciences: Spectral Asymptotics of Toeplitz and Pseudodifferential Operators

这项工作继续数学研究到 某些微分的渐近展开式， 伪微分算子 当一个微分变换 通过经典变换，该算子通常 显示为变换后的解的乘数。 这 乘数成为运算符的符号。 渐近 光滑和非光滑符号的扩展已经在 不同的背景。 进一步发展的一个障碍是， 一类非光滑积分驻相法缺乏 振幅函数 最近的事态发展使人们可以得出 某些类别的不连续符号的完全扩展。 这项工作的目的是继续朝这个方向努力， 接手更困难更有趣的案子 最终 人们将寻找一个精确的一般理论。

This work continues mathematical research into the asymptotic expansions of certain differential and pseudodifferential operators. When one transforms a differential operator by means of a classical transform, the operator often appears as a multiplier of the transformed solution. This multiplier becomes the symbol of the operator. The asymptotic expansion of smooth and nonsmooth symbols has been obtained in different contexts. One barrier to going further has been the lack of method of stationary phase for integrls with a nonsmooth amplitude fundtion. Recent developments now allow one to derive complete expansions for certain classes of discontinuous symbols. It is the purpose of this work to continue in this direction, taking on more difficult and more interesting cases. Ultimately one will look for a precise general theory.