Spectral analysis on interior transmission eigenvalues

In this paper we prove some results on interior transmission eigenvalues. First, under rea- sonable assumptions, we prove that the spectrum is a discrete countable set and the generalized eigenfunctions spanned a dense space in the range of resolvent. This is a consequence of spectral theory of Hilbert-Schmidt operators. The main ingredient is to prove a smoothing property of resolvent. This allows to prove that a power of the resolvent is Hilbert-Schmidt. We obtain an estimate of the number of eigenvalues, counting with multiplicities, with modulus less than t2 when t is large. We prove also some estimate on the resolvent near the real axe when the square of the index of refraction is not real. Under some assumptions we obtain lower bound on the resolvent using the results obtained by Dencker, Sj\"ostrand and Zworski on the pseudospectra.