A review of the results on second analytic singularities in diffraction problems (Several aspects of microlocal analysis)

Abstract

We survey the microlocal singularity theory of diffraction problems in the analytic category, in particular, J. Sjöstrand s theory [8, 9, 10] on propagation of micro-analyticity along the boundary, and G. Lebeau s celebrated theory on the construction of the parametrix in [3]. The title of [3] looks like a paper on Gevrey singularities, but his parametrix is constructed in the second analytic category, that is, modulo distributions with one holomorphic parameter. So, we can conclude that the boundary values of the diffraction wave have second analyticity in the shadow. We also give such a parametrix for a typical example of diffraction problems with some well-known properties of Airy integrals.