Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems

Abstract

We consider several intriguingly connected topics in the theory of wave propagation: geometrical characterizations of radiationless sources, non-radiating incident waves, interior transmission eigenfunctions, and their applications to inverse scattering. Our major novel discovery is a localization and geometrization property. We first show that a scatterer, which might be an active source or an inhomogeneous index of refraction, cannot be completely invisible if its support is small compared to the wavelength and scattering intensity. Next, we localize and geometrize the "smallness" results to the case where there is a high-curvature point on the boundary of the scatterer's support. We derive explicit bounds between the intensity of an invisible scatterer and its diameter or its curvature at the aforementioned point. These results can be used to characterize radiationless sources or non-radiating waves near high-curvature points. As significant applications we derive new intrinsic geometric properties of interior transmission eigenfunctions near high-curvature points. This is of independent interest in spectral theory. We further establish unique determination results for the single-wave Schiffer's problem in certain scenarios of practical interest, such as collections of well-separated small scatterers. These are the first results for Schiffer's problem with generic smooth scatterers.

arXiv comment35 pages, 1 figure

References in corpus (12)

• Spectral analysis on interior transmission eigenvalues
• On vanishing near corners of transmission eigenfunctions
• Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements
• Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems
• Reconstruction of a source domain from the Cauchy data
• Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem
• Regularized Transformation-Optics Cloaking for the Helmholtz Equation: From Partial Cloak to Full Cloak
• Addendum to: "On vanishing near corners of transmission eigenfunctions"
• Regularized Transformation-Optics Cloaking in Acoustic and Electromagnetic Scattering
• Translation-Invariant Estimates for Operators with Simple Characteristics
• Recovering piecewise constant refractive indices by a single far-field pattern
• Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions

Cited by in corpus (8)

• On an electromagnetic problem in a corner and its applications
• On a local geometric property of the generalized elastic transmission eigenfunctions and application
• On Calderón's inverse inclusion problem with smooth shapes by a single partial boundary measurement
• On Corner Scattering for Operators of Divergence Form and Applications to Inverse Scattering
• On local and global structures of transmission eigenfunctions and beyond
• A New Type of CGO Solutions and its Applications in Corner Scattering
• On new surface-localized transmission eigenmodes
• Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes