Stationary phase and microlocal analysis

Abstract

This chapter discusses basic techniques from the theory of stationary phase. After giving an overview of the method of stationary phase, the chapter moves on to a discussion of pseudodifferential operators, by going over the basics from the calculus of pseudodifferential operators and their various microlocal properties, in the process obtaining an equivalent definition of wave front sets, before defining pseudodifferential operators on manifolds and going over some of their properties. The chapter then lays out the propagation of singularities as well as Egorov's theorem, which involves conjugating pseudodifferential operators. Finally, this chapter describes the Friedrichs quantization, and differentiates it from the Kohn-Nirenberg quantization presented earlier in the chapter.