Transmission eigenvalue-free regions near the real axis. II

TL;DR

This paper extends previous results to more general domains, establishing transmission eigenvalue-free regions for problems with complex refraction indices and damping, without requiring non-trapping or strict concavity conditions.

Contribution

It introduces new conditions based on high-frequency resolvent behavior, removing previous geometric restrictions such as non-trapping and strict concavity.

Findings

Established eigenvalue-free regions for complex refraction indices

Removed non-trapping and strict concavity assumptions

Utilized high-frequency resolvent estimates for general domains

Abstract

In this paper we extend the results in [16] to more general domains. More precisely, we obtain transmission eigenvalue-free regions for the interior transmission problem with one complex-valued refraction index, that is, with a damping term which does not vanish on the boundary. In particular, we remove the non-trapping condition as well as the strict concavity condition from [16]. Instead, we impose new, more general conditions in terms of the highfrequency behavior of certain cut-off resolvents associated to exterior problems.