Spectral analysis of the interior transmission eigenvalue problem

In this paper we prove some results on interior transmission eigenvalues. First, under reasonable assumptions, we prove that the spectrum is a discrete countable set and that the generalized eigenfunctions span a dense subspace in the range of resolvent. This is a consequence of the spectral theory of Hilbert–Schmidt operators. The main ingredient is the proof of a smoothing property of the resolvent. This allows us to prove that a power of the resolvent is Hilbert–Schmidt. We obtain an estimate of the number of eigenvalues, counted with multiplicities, with modulus less than t2 when t is large. We also prove some estimate on the resolvent near the real axis when the square of the refraction index is not real. Under some assumptions we obtain a lower bound for the resolvent using results obtained by Dencker, Sjostrand and Zworski on the pseudospectrum.