Interior transmission eigenvalue problems on compact manifolds with boundary conductivity parameters

TL;DR

This paper investigates interior transmission eigenvalues on compact Riemannian manifolds with boundary, establishing their discreteness, infinite existence, and Weyl bounds, with applications to non-scattering energies in acoustic scattering.

Contribution

It extends the analysis of ITE problems to manifolds with boundary conductivity, proving key spectral properties and applying the results to scattering theory.

Findings

Set of ITEs is discrete

Existence of infinitely many ITEs

Weyl type lower bound for ITEs

Abstract

In this paper, we consider an interior transmission eigenvalue (ITE) problem on some compact $C^{\infty }$-Riemannian manifolds with a common smooth boundary. In particular, these manifolds may have different topologies, but we impose some conditions of Riemannian metrics, indices of refraction and boundary conductivity parameters on the boundary. Then we prove the discreteness of the set of ITEs, the existence of infinitely many ITEs, and its Weyl type lower bound. For our settings, we can adopt the argument by Lakshtanov and Vainberg, considering the Dirichlet-to-Neumann map. As an application, we derive the existence of non-scattering energies for time-harmonic acoustic equations. For the sake of simplicity, we consider the scattering theory on the Euclidean space. However, the argument is applicable for certain kinds of non-compact manifolds with ends on which we can define the scattering matrix.