We prove the existence of large regions free of eigenvalues of the interior transmission problem.Then there are no transmission eigenvalues in Λ − ∪ Λ + , whereRemark 1. In the case c 1 ≡ c 2 ≡ 1 it was proved previously in [7] that outside any sector |Im λ| ≤ θRe λ, ∀θ > 0, there are at most a finite number of transmission eigenvalues. In the

We prove uniform weighted high frequency estimates for the resolvent of the Laplace-Beltrami operator on connected infinite volume Riemannian manifolds under some natural assumptions on the metric on the ends of the manifold. This extends previous results by Burq [3] and Vodev [8].

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