Localization of the interior transmission eigenvalues for a ball

Abstract

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball {x ∈ R d : |x| ≤ 1}, d ≥ 2, and the coefficients cj (x), j = 1, 2, and the indices of refraction nj (x), j = 1, 2, are constants near the boundary |x| = 1. We prove that in this case the eigenvalue-free region obtained in [17] for strictly concave domains can be significantly improved. In particular, if cj (x), nj (x), j = 1, 2 are constants for |x| ≤ 1, we show that all (ITEs) lie in a strip |Im λ| ≤ C.

A duality between scattering poles and transmission eigenvalues in scattering theory

“…Note that by separating variables in (1.7), we can see that all transmission eigenvalues for the spherically symmetric media are obtained from C ℓ ( k ; n ) = 0 for ℓ∈double-struckN. The transmission eigenvalues for spherically symmetric cases are extensively studied in [14–17]. In particular, it is shown that (except for some exceptional cases) the entire functions C ℓ ( k ; n ) have infinitely many real zeros and infinitely many complex zeros.…”

“…In this case, every transmission eigenvalue is a non-scattering wave number, since by construction at a transmission eigenvalue, the eigenfunctions of (1.7) with D := B 1 (0) and n := n(r) are linear combinations of v = j (k|x|)Y (x) and u := v + u s with u s given by (1.8). Note that by separating variables in (1.7), we can see that all transmission eigenvalues for the spherically symmetric media are obtained from C (k; n) = 0 for ∈ N. The transmission eigenvalues for spherically symmetric cases are extensively studied in [14][15][16][17]. In particular, it is shown that (except for some exceptional cases) the entire functions C (k; n) have infinitely many real zeros and infinitely many complex zeros.…”

A note on transmission eigenvalues in electromagnetic scattering theory

“…The best to date result for Maxwell's equations is presented in [8], where more precisely it is shown that transmission eigenvalues lie inside any arbitrary small wedge around the real and imaginary axis. In contrast, for the scalar case of Helmholtz equation, much finer results on the location of transmission eigenvalues are available [15,20,21,22,23]. More importantly, for the Helmholtz equation, Vodev [23] is the first to show that for inhomogeneities with smooth support and coefficients, there are no transmission eigenvalues outside a horizontal strip around the real axis, i.e.…”

“…We recall that this is a non-selfadjoint eigenvalue problem and complex eigenvalues are proven to exist for spherically symmetric media [10,11,18], and for some cases of general media in [21]. The location of transmission eigenvalues of (9), which is the main concern for us, has been studied in [20,21,22,23] for C ∞ -smooth media. We are particularly interested in media for which the imaginary part of transmission eigenvalues remain bounded.…”