On certain translation invariant properties of interior transmission spectra and their Doppler's effect (Q1798399)

Summary: We study the translation invariant properties of the eigenvalues of scattering transmission problem. We examine the functional derivative of the eigenvalue density function \(\Delta(\hat{x})\) to the defining index of refraction \(n(x)\). By the limit behaviors in frequency sphere, we prove some results on the inverse uniqueness of index of refraction. In physics, Doppler's effect connects the variation of the frequency/eigenvalue and the motion velocity/variation of position variable. In this paper, we proved the functional derivative \(\partial_r \Delta \left(\hat{x}\right) = (1 + \sqrt{n \left(r \left(\hat{x}\right)\right)}) / \pi\).