Mathematical Sciences: Spectral Asymptotics of Toeplitz and Pseudodifferential Operators

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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.