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Inside-outside duality and time-harmonic wave scattering

There is an interesting relation in time-harmonic inverse scattering theory between the interior eigenvalues of certain scattering objects and the spectrum of the corresponding measurement operators. The eigenvalues of the measurement operator always tend to zero due to compactness. In several cases, one can prove that they tend to zero either from the left or from the right in the complex plane. In this situation, the inside-outside duality says that whenever some eigenvalue tends to zero from the ’wrong’ side as the wave number tends to some value k_0 if and only if k_0 is an interior eigenvalue. We show how this relation arises and detail several consequences, e.g., the detection of interior eigenvalues from far field data and a monotonicity principle for inverse medium scattering.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46