Weyl asymptotics of the transmission eigenvalues for a constant index of refraction

Abstract

We prove Weyl type of asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.

Localization of the interior transmission eigenvalues for a ball

“…(A) Prove the discreteness of the spectrum of A in C; (B) Find eigenvalue-free regions in C; (C) Establish a Weyl formula for the counting function of all (ITEs) N (r) = #{λ j is (ITE), |λ j | ≤ r}. 1 Note that the problem (A) is now relatively well studied (see [9], [14], [11], [4] and the references therein). In fact, the problem (A) is reduced to that one of showing that the resolvent of A is meromorphic with residues of finite rank.…”

“…In the isotropic case when c j ≡ 1, j = 1, 2 and n 1 = 1, n 2 = 1 is constant, the eigenvalue-free region (1.7) has been established in the one-dimensional case Ω = {x ∈ R : |x| ≤ 1} (see [14], [11]). Moreover, the case of the ball {x ∈ R d : |x| ≤ 1}, d = 2, 3, and radial refraction indices have been studied in [9], [3], [4], where spherical symmetric eigenfunctions depending only on the radial variable r = |x| has been considered.…”

Location and Weyl Formula for the Eigenvalues of Some Non Self-Adjoint Operators

“…This leads to the following improvement of Theorem 4. This results is almost optimal, since for the unit ball in R d we have the following The case d = 1 and K = {x ∈ R : |x| ≤ 1} has been previously examined in [18] and [16].…”

“…• The optimal result should be to have a eigenvalues-free region with κ = 1 as it was proved in [15], [18], [16] for the case when K is a ball and the functions c j , n j are constants. However, even with κ = 1, to obtain an optimal remainder O(r d−1 ) some extra work is needed and this is an interesting open problem.…”

High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues

“…We also expect that (2.9) holds with ε = 0, but this remains an interesting open problem. In the isotropic case asymptotics for the counting function N (r) with remainder o(r d ) have been previously obtained in [4], [8], [13].…”