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

Abstract

Abstract. We study the high-frequency behavior of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use this fact to get new results concerning the location of the transmission eigenvalues on the complex plane. In some cases we obtain optimal transmission eigenvalue-free regions.

New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

“…We obtain here a conditional monotonicity property for the largest positive Steklov eigenvalue. In the following theorem we give the optimal condition on A, n and k which ensure the coercivity property (25), whence the sup-condition (27). (20).…”

Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann map

“…Having high-frequency approximations of the DN map proves very usefull when studying the location of the complex eigenvalues associated to boundary value problems with dissipative boundary conditions or to interior transmission problems. In particular, this proves crucial to get parabolic transmission eigenvalue-free regions (see [9], [10], [11], [12]). As an application of our parametrix we improve the transmission eigenvalue-free region obtained in [12] in the case of the degenerate isotropic interior transmission problem (see Theorem 4.1).…”

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.…”

“…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. all transmission eigenvalues have imaginary part uniformly bounded, under certain conditions for the contrasts of the medium.…”

“…This short note addresses the location of electromagnetic transmission eigenvalues in the complex plane motivated by desire to eventually study the solvability of time dependent interior transmission problem for Maxwell's equations. In the process of trying to find whether there are possible combination of , 0 and µ, µ 0 for which transmission eigenvalues lie in a strip, we noticed that there are no conditions on these coefficients that can bring the transmission eigenvalue problem to the form for which one can adapt high frequency estimates of Vodev in [23], which allow for showing that the imaginary part of transmission eigenvalues is uniformly bounded. More specifically, restricting ourselves to spherically symmetric isotropic media, we show that the set of transmission eigenvalues for Maxwell's equations can be divided into two subsets, each corresponding to a transmission eigenvalue problem for Helmholtz equations.…”