Conditions against rapid decrease of oscillatory integrals and their applications to inverse scattering problems
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.
scientific article; zbMATH DE number 4000572