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 $\mathcal{A}_2$ denotes the Hilbert space operator induced by this boundary problem, then in order to derive information concerning the spectral properties of $\mathcal{A}_2$, 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 $\mathcal{A}_2$ is just a linearization of $V_2(\lambda)$, we thus arrive at the spectral properties of $\mathcal{A}_2$. Finally, by appealing to some known results pertaining to the uniqueness ...

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