Geometric Properties of Helmholtz’s Transmission Eigenfunctions Induced by Curvatures and Applications

In this chapter, we consider two types of scenarios in inverse scattering problems, where we follow the treatment in Blåsten et al. (SIAM J Math Analy 53(4), 3801–3837, 2021). The first one is concerned with radiationless or non-radiating monochromatic sources, and the other one is concerned with non-radiating waves that impinge against a certain given scatterer consisting of an inhomogeneous index of refraction. In this chapter, we show that such invisible objects have certain geometrical properties. This allows us to classify radiating sources and incident waves that are always radiating for scatterers. Moreover, they also help us to provide unique determination results for a longstanding inverse scattering problem in certain scenarios of practical importance. The second one is concerned with geometrical structure of transmission eigenfunctions at highly curved point. First, we provide a relationship among the value of the transmission eigenfunction, the diameter of the domain, and the underlying refractive index, which indicates that if the domain is sufficiently small, then the transmission eigenfunction is nearly vanishing. Then we further localize and geometrize this “smallness” result. Briefly, interior transmission eigenfunctions must be nearly vanishing at a high-curvature point on the boundary. Moreover, the higher the curvature, the smaller the eigenfunction must be at the high-curvature point. This nearly vanishing behavior readily implies that as long as the shape of a scatterer possesses a highly curved part, then it scatters every incident wave field nontrivially unless the wave is vanishingly small at the highly curved part. Furthermore, we give uniqueness results for Schiffer’s problem in determining the support of an active source or an inhomogeneous medium with a single far-field pattern in certain scenarios of practical interest. These are the unique determination results for Schiffer’s problem concerning scatterers with smooth shapes. The corresponding uniqueness result on Calderón’s inverse inclusion problem with smooth shapes by a single partial boundary measurement can be found in Liu et al. (On Calderón’s inverse inclusion problem with smooth shapes by a single partial boundary measurement, arXiv:2006.10586).