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An inverse spectral uniqueness in exterior transmission problem. (English) Zbl 1390.35197
The paper is concerned with the scattering problem \[ \begin{cases} \Delta u(x)+k^2n(x)u(x)=0,\,\,\,x\in\mathbb{R}^3,\\ u(x)=u^i(x)+u^s(x),\,\,\,x\in\mathbb{R}^3\setminus D,\\ \lim_{|x|\to\infty}|x|\left\{\frac{\partial u^s(x)}{\partial|x|}-iku^s(x)\right\}=0, \end{cases}(1) \] where \(u(x)\) is the total wave, \(u^s(x)\) is the scattered wave, \(u^i(x)=e^{ikx\cdot d}\) is the incident wave, \(k\in \mathbb{C}\), \(x\in \mathbb{R}^3\), \(d\in \mathbb{S}^2\), \(D\) is a starlike domain in \(\mathbb{R}^3\) which contains the origin with the boundary \(\partial D\), \(\text{supp}\,(1-n)\) is outside \(D\), simple and contained in a bounded domain \(D'\). Here the inhomogeneity \(n\in {\mathcal C}^2(\mathbb{R}^3)\), \(n(x)>0\) for all \(x\in \mathbb{R}^3\), and the Laplacian \(\Delta\) is defined by \[ \Delta=\frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\partial}{\partial r}+\frac{1}{r^2 \sin\varphi}\frac{\partial}{\partial\varphi}\sin\varphi\frac{\partial}{\partial \varphi}+\frac{1}{r^2\sin^2\varphi}\frac{\partial^2}{\partial\theta^2}. \] The author investigates the inhomogeneities to the background index of refraction \(1\) to the far-fields for the inverse eigenvalue problem associated to \((1)\).
MSC:
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P25 Scattering theory for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35R30 Inverse problems for PDEs
34B24 Sturm-Liouville theory
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