THE INTERIOR TRANSMISSION PROBLEM: SPECTRAL THEORY

In this paper we are concerned with the interior transmission eigenvalue problem connected with a degenerate boundary problem with limited smoothness assumptions concerning its coefficients and boundary. If A(2) denotes the Hilbert space operator induced by this boundary problem, then in order to derive information concerning the spectral properties of A(2), we are led to consider an auxiliary boundary problem involving powers of the spectral parameter lambda up to the second order. Under our assumptions we show that the auxiliary boundary problem is parameter-elliptic, and hence we can now appeal to the theory concerning such problems to derive information pertaining to the spectral properties of the quadratic operator pencil V-2(lambda) induced by the auxiliary boundary problem. Since A(2) is just a linearization of V-2(lambda), we thus arrive at the spectral properties of A(2). Finally, by appealing to some known results pertaining to the uniqueness of the Cauchy problem, we show that under possibly some further conditions, the spectral theory derived for A(2) is precisely that for the interior transmission problem under consideration.