The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions

Interior transmission problems with coefficients of low regularity

“…It follows from the results in [9] that under the above conditions the transmission eigenvalues are discrete in both cases, so it is natural to ask if the counting function of these transmission eigenvalues admits Weyl asymptotics. Indeed, such asymptotics were obtained in [11], [12] in the case of C ∞ smooth coefficients, and more recently in [4], [10] for coefficients of very low regularity but with much worse remainder terms. The proof in [11] relies heavily on the location of the transmission eigenvalues on the complex plane and the C ∞ regularity of the coefficients is not essential.…”

Transmission Eigenvalues

“…Hence, using real transmission eigenvalues to detect changes in anisotropic media becomes quite important (see [CCH16] and the references therein). More generally, completeness results on transmission eigenfunctions and Weyl's estimates on the counting function are obtained in [NN21] for and continuous in a neighborhood of satisfying the following two conditions: ⟨ ( ) , ⟩ ⟨ ( ) , ⟩ − ⟨ ( ) , ⟩ ≠ 1 ∀ ∈ and for every ∈ ℝ 3 ⧵ {0} orthogonal to the outward normal vector at any ∈ , and ⟨ ( ) , ⟩ ( ) ≠ 1 for all ∈ (the first condition is known as the complementing condition due to Agmon, Douglis, and Nirenberg). Under ∞ regularity for both the boundary and the coefficients and assuming that ( ) = ( ) , where is a scalar function, the location in the complex plane of the transmission eigenvalues is studied in [Vod18].…”

Semiclassical parametrix for the Maxwell equation and applications to the electromagnetic transmission eigenvalues

“…This question has been largely investigated in the context of the acoustic transmission eigenvalues, that is, those associated to the Helmholtz equation. Several sufficient condition have been found that guarantee not only the discreteness, but also Weyl asymptotics for the counting function of the acoustic transmission eigenvalues (see [5], [6], [7]). In particular, it was proved in [6] that the existence of parabolic eigenvalue-free regions implies the Weyl asymptotics.…”