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Propagation of polyhomogeneity, Diffraction and Scattering on Product Cones

Mengxuan Yang

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-Taylor's classical result of half-step polyhomogeneity of diffractive waves in [CT82a], [CT82b]. 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 [FHH18]. In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.

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-Taylor's classical result of half-step polyhomogeneity of diffractive waves in [CT82a], [CT82b]. 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 [FHH18]. In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.

Paper Structure

This paper contains 6 sections, 16 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conic Diffraction Geometry
  3. Propagation of Polyhomogeneity
  4. Diffraction Coefficient on Product Cones
  5. Scattering Matrix on Product Cones
  6. From the Radiation Field to the Scattering Matrix

Key Result

Theorem 1.1

Away from the intersection of the geometric wave front and diffractive wave front, the kernel of the diffractive half wave propagator is a conormal distribution of the form: The principal symbol $K_0(r,r',\theta,\theta')$ of the diffractive half wave kernel $U_{D}(t,r,r',\theta,\theta')$ is related to the kernel of the scattering matrix $S(\lambda, \theta,\theta')$ by

Figures (2)

  • Figure 1: Diffractive and geometric geodesics
  • Figure 2: Geometric and Diffractive Front

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.1.1
  • Definition 2.1
  • Definition 3.1: Polyhomogeneous symbols
  • Definition 3.2: Conormality on cones
  • Definition 3.3: Conormality again
  • Remark 3.1
  • Theorem 3.1
  • ...and 21 more