Surface Concentration of Transmission Eigenfunctions

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in Chow et al. (SIAM J Imaging Sci, 14(3):946–975, 2021) on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based a generalized Weyl’s law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.