An eigenvalue associated with the interior transmission problem: a value of the spectral parameter for which there exist nontrivial interior fields on either side of an interface that satisfy the transmission continuity conditions (field and flux matching) across the interface in the absence of incident waves.

Principle

Transmission eigenvalues arise from coupling two PDEs across an interface via continuity of traces and conormal derivatives; they characterize frequencies at which the homogeneous transmission system admits nontrivial interior solutions without external forcing.

Demonstration

In acoustic scattering with a bounded inclusion of different refractive index, a transmission eigenvalue k^2 is a wave number for which there exist nonzero interior pressure fields u and v solving Helmholtz equations with different coefficients inside and outside the inclusion and satisfying u = v and ∂_ν u = ∂_ν v on the interface, with no incoming field.

Misapplication

Treating transmission eigenvalues as ordinary Dirichlet or Neumann eigenvalues of a single domain; unlike classical boundary eigenvalues, transmission eigenvalues stem from a coupled interior–interior problem and may not correspond to a single self-adjoint operator spectrum.

Consequence

Transmission eigenvalues influence inverse scattering: their existence and distribution carry information about material contrasts and can be used to reconstruct or constrain interior properties of the scatterer; they also signal frequencies at which interior non-uniqueness of the homogeneous transmission system occurs.

Reversal

A scattering resonance or pole of the exterior scattering operator is the inverse notion: resonances relate to outgoing solutions in the exterior domain with no incident field, while transmission eigenvalues involve interior nontrivial paired solutions satisfying exact matching across the interface.

Boundary

Defined for problems with a bounded interface separating media with differing coefficients and sufficient regularity; excluded are strongly absorbing media (complex indices with large imaginary parts) where the interior transmission formulation must be modified, and situations without a clear two-region decomposition.

Semantic Tension

There is tension between transmission eigenvalues and interior Dirichlet/Neumann spectra or scattering resonances: they share spectral flavor but differ in origin (coupled interior problem vs single-domain eigenvalue or exterior pole), so naive identification among these leads to errors.

Synthesis

A transmission eigenvalue is a spectral parameter at which the homogeneous interior transmission problem admits nontrivial matched interior fields across an interface; it encodes material contrast information distinct from classical boundary eigenvalues and from exterior scattering resonances.