An inverse uniqueness on the index of refraction with a transition region

A Proof of Schiffer's Conjecture in Starlike Domain by Far-Field Patterns

“…Here we prove the result as a special case of interior transmission problem [2,4,10,11,12,13,14,15,16,17,18,21,22,24,25,28,29,31]. In interior transmission problems, we look for a frequency so that a perturbed stationary wave behaves like or somewhere like a spherical Bessel function outside the perturbation.…”

A fixed energy fixed angle inverse scattering in interior transmission problem

“…Most important of all, the estimates from (2.14) to (2.17) show that they are entire functions of order one and of type ξ. We refer details to [1,10,12,14,23,25]. Accordingly their zero sets are described by Cartwright's theory [2,8,9,21,22].…”

“…These two terms do not cancel each other if n 1 = n 2 . We skip the details and refer it to [10,11,12,15].…”

“…where ν is the unit outer normal. The equation (1.4) is called the homogeneous interior transmission eigenvalue problem [1,3,4,10,12,13,14,16,18,20,23,26,28]. The problem (1.4) occurs naturally when one considers the scattering of the plane waves by certain inhomogeneity inside the domain Ω, defined by an index of refraction in many models.…”

An Inverse Uniqueness of a Phaseless Scattering Problem by Zero-Crossings

“…Whenever the zero crossing density exceeds its theoretical maximal quantity, the function must be identically zero. This renders an inverse uniqueness on the scatterer, as we have discussed in [3,4,5]. The assumption on the neighborhood U in Theorem 1.1 can be replaced by the assumptions on the numerical quantity of the zero-crossing density of f (x, ν, k).…”