An inverse spectral uniqueness in exterior transmission problem (Q2413073)

The paper is concerned with the scattering problem \[ \begin{cases} \Delta u(x)+k^2n(x)u(x)=0,\,\,\,x\in\mathbb{R}^3,\\ u(x)=u^i(x)+u^s(x),\,\,\,x\in\mathbb{R}^3\setminus D,\\ \lim_{|x|\to\infty}|x|\left\{\frac{\partial u^s(x)}{\partial|x|}-iku^s(x)\right\}=0, \end{cases}\leqno(1) \] 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 \((1)\).