Propagation of polyhomogeneity, diffraction, and scattering on product cones

Abstract

We consider diffraction of waves on a product cone. We first show that diffractive waves enjoy a one-step polyhomogeneous asymptotic expansion, which is an improvement of Cheeger and Taylor's classical result of half-step polyhomogeneity of diffractive waves [Comm. Pure Appl. Math. 35 (1982), 275-331 and 487-529]. We also conclude that on product cones, the scattering matrix is the diffraction coefficient, which is the principal symbol of the diffractive half wave kernel, for strictly diffractively related points on the cross section. This generalize the result of Ford, Hassell and Hillairet in 2-dimensional flat cone settings (Ford, Hassell, and Hillairet, 2018). In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.

Resolvents and complex powers of semiclassical cone operators

“…Such estimates have been obtained in a number of different settings: Cheeger–Taylor [5, 6], see also [21], analyzed wave propagation on exact cones, followed by Melrose–Wunsch [37], using geometric microlocal techniques, on general conic manifolds, see also [36, 46]; these papers also prove diffractive regularity improvements via the use of the edge calculus [29, 31]. (For recent results on the fine properties of the diffracted wave, see [9, 49]. ) More recently, Baskin–Marzuola [3] proved semiclassical propagation estimates on unweighted spaces (relative to the quadratic form domain).…”