Conditions against rapid decrease of oscillatory integrals and their applications to inverse scattering problems
type: Item
instance of: scholarly article
title: Conditions against rapid decrease of oscillatory integrals and their applications to inverse scattering problems
zbMATH Open document ID: 0617.35104
author: Hideo Soga
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The author applies results on oscillatory integrals to two problems, the first to do with scattering by a bounded obstacle \(O\subset {\mathbb{R}}^ n\), \(n\geq 2\) and the second with scattering through an inhomogeneous medium. These are given by \[ \partial^ 2_ tu- \sum^{n}_{i,j=1}\partial_{x_ i}(a_{ij}(x)\partial_{x_ j}u)=0\text{ in } {\mathbb{R}}\times \Omega, \] \[ (*)\quad u(t,x)=0\text{ on } {\mathbb{R}}\times \partial \Omega,\quad u(0,x)=f_ 1(x),\quad \partial_ tu(0,x)=f_ 2(x)\text{ on } \Omega. \] In the first problem \(\Omega ={\mathbb{R}}^ n\setminus O\), \(\partial \Omega =\partial O\) and \(a_{ij}(x)=\delta_{ij}\), while in the second, \(\Omega ={\mathbb{R}}^ n\), (*) does not apply and the matrix \(a_{ij}(x)\) is real, symmetric, \(C^{\infty}\), and positive definite. Outside a ball \(| x| >r_ 0\), \(a_{ij}(x)\) reduces again to \(\delta_{ij}.\) If \(k_{\pm}(s,\omega)\in L^ 2({\mathbb{R}}\times S^{n-1})\) denote the translation representations for the data \(f_ 1,f_ 2\), and \(S: k_- \to k_+\) is the scattering operator represented through a scattering kernel \(S(s,\theta,\omega)\), it is shown in the first problem that for fixed \(\omega,\theta \in S^{n-1}\), \(\omega \neq \theta\), (i) supp \(S(.,\theta,\omega)\subset (-\infty,-r(\omega -\theta)]\), and (ii) \(S(s,\theta,\omega)\) is singular at \(s=-r(\omega -\theta)\) where \(r(\phi)=\min_{x\in O} x\cdot \phi\). Corresponding results are obtained for the second problem. The title of the paper comes from the analysis required on the Fourier integral operators used (and developd by means of an associated problem) in the construction of the scattering kernel.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 4000572
zbMATH Keywords: integral representation
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: scattering
zbMATH Keywords: bounded obstacle
zbMATH Keywords: inhomogeneous medium
zbMATH Keywords: scattering kernel
reviewed by: Ralph Saxton
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4000572