Mathematical Sciences: Spectral Asymptotics of Toeplitz and Pseudodifferential Operators

这项工作继续对某些微分算子和伪微分算子的渐近展开进行数学研究。当用经典变换变换微分算子时，该算子通常表现为变换后解的乘子。这个乘数成为运算符的符号。得到了光滑和非光滑符号在不同情况下的渐近展开式。对于具有非光滑振幅基础的积分，缺乏固定相位的方法是进一步研究的一个障碍。最近的发展现在允许人们推导出某些不连续符号的完全展开式。这项工作的目的是沿着这个方向继续下去，承担更困难和更有趣的案例。最终，人们将寻求一种精确的一般理论。

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.