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In this paper, we consider an interior transmission eigenvalue (ITE) problemon some compact $C^{\infty }$-Riemannian manifolds with a common smoothboundary. In particular, these manifolds may have different topologies, but weimpose some conditions of Riemannian metrics, indices of refraction andboundary conductivity parameters on the boundary. Then we prove thediscreteness of the set of ITEs, the existence of infinitely many ITEs, and itsWeyl type lower bound. For our settings, we can adopt the argument byLakshtanov and Vainberg, considering the Dirichlet-to-Neumann map. As anapplication, we derive the existence of non-scattering energies fortime-harmonic acoustic equations. For the sake of simplicity, we consider thescattering theory on the Euclidean space. However, the argument is applicablefor certain kinds of non-compact manifolds with ends on which we can define thescattering matrix.

Interior transmission eigenvalue problems on compact manifolds with boundary conductivity parameters