Abstract
Volume integral equations have been used as theoretical and numerical tools in scattering theory for a long time. The basic idea of the VIE method in scattering by a penetrable object is to consider the effect of the scatterer as a perturbation of a whole-space constant coefficient problem and to solve the latter by convolution with the whole-space fundamental solution. In acoustic and electromagnetic scattering, this results in strongly singular integral equations that have a non-trivial essential spectrum, in general. Using techniques developed for the case of electromagnetic scattering (see the recent note by M. Costabel, E. Darrigrand and H. Sakly, The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body. Comptes Rendus Mathématique, 350, 193–197 (2012)), we determine the essential spectrum, hence the well-posedness in the sense of Fredholm, of the volume integral equation in acoustic scattering. It turns out that the question can be reduced to the study of the classical double layer potential boundary integral operator.
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Costabel, M. (2015). On the Spectrum of Volume Integral Operators in Acoustic Scattering. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_11
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DOI: https://doi.org/10.1007/978-3-319-16727-5_11
Publisher Name: Birkhäuser, Cham
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