  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 [10] on such an intriguing
spectral phenomenon of the transmission resonance. Our argument is based on
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.
