We consider several intriguingly connected topics in the theory of wave propagation: geometrical characterizations of radiationless sources, nonradiating 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 nonradiating 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.


  1. radiationless sources
  2. invisible
  3. transmission eigenfunctions
  4. inverse shape problems
  5. geometrical properties
  6. single far-field pattern

MSC codes

  1. Primary
  2. 35Q60
  3. 78A46; Secondary
  4. 35P25
  5. 78A05
  6. 81U40

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3801 - 3837
ISSN (online): 1095-7154


Submitted: 2 December 2020
Accepted: 29 March 2021
Published online: 12 July 2021


  1. radiationless sources
  2. invisible
  3. transmission eigenfunctions
  4. inverse shape problems
  5. geometrical properties
  6. single far-field pattern

MSC codes

  1. Primary
  2. 35Q60
  3. 78A46; Secondary
  4. 35P25
  5. 78A05
  6. 81U40



Funding Information

Hong Kong Research Grants Council : 2302017, 12301218, 12302919
Academy of Finland https://doi.org/10.13039/501100002341 : 312124
Eesti Teadusagentuur https://doi.org/10.13039/501100002301 : PRG 832

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