Recovering two coefficients in an elliptic equation via phaseless information

For fixed $y \in \mathbb{R}^3$ , we consider the equation $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$ , where $L=\text{div}(n(x)^{-2}\nabla)+q(x)$ , $k >0$ is a frequency, $n(x)$ is a refraction index and $q(x)$ is a potential. Assuming that the refraction index $n(x)$ is different from $1$ only inside a bounded compact domain $Ω$ with a smooth boundary $S$ and the potential $q(x)$ vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside $Ω$ from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point $y \in S$ is a variable parameter of the problem. Then for the solution $u(x,y,k)$ of the above equation satisfying the radiation condition, we assume to be given the following phaseless information $f(x,y,k)=|u(x,y,k)|^2$ for all $x,y \in S$ and for all $k≥ k_0>0$ , where $k_0$ is some constant. We prove that this phaseless information uniquely determines both coefficients $n(x)$ and $q(x)$ inside $Ω$ .

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