On the Spectrum of Volume Integral Operators in Acoustic Scattering

TL;DR

This paper investigates the spectral properties and well-posedness of volume integral operators in acoustic scattering, extending the mathematical understanding of these operators beyond electromagnetic cases.

Contribution

It provides a spectral analysis of volume integral operators in acoustic scattering and establishes conditions for their Fredholm properties, filling a gap in the mathematical literature.

Findings

Spectral analysis of acoustic volume integral operators.

Necessary and sufficient conditions for Fredholm well-posedness.

Extension of electromagnetic results to acoustic scattering.

Abstract

Volume integral equations have been used as a theoretical tool in scattering theory for a long time. A classical application is an existence proof for the scattering problem based on the theory of Fredholm integral equations. This approach is described for acoustic and electromagnetic scattering in the books by Colton and Kress [CoKr83, CoKr98] where volume integral equations appear under the name "Lippmann-Schwinger equations". In electromagnetic scattering by penetrable objects, the volume integral equation (VIE) method has also been used for numerical computations. In particular the class of discretization methods known as "discrete dipole approximation" [PuPe73, DrFl94] has become a standard tool in computational optics applied to atmospheric sciences, astrophysics and recently to nano-science under the keyword "optical tweezers", see the survey article [YuHo07] and the literature quoted there. In sharp contrast to the abundance of articles by physicists describing and analyzing applications of the VIE method, the mathematical literature on the subject consists only of a few articles. An early spectral analysis of a VIE for magnetic problems was given in [FrPa84], and more recently [Ki07, KiLe09] have found sufficient conditions for well-posedness of the VIE in electromagnetic and acoustic scattering with variable coefficients. In [CoDK10, CoDS12], we investigated the essential spectrum of the VIE in electromagnetic scattering under general conditions on the complex-valued coefficients, finding necessary and sufficient conditions for well-posedness in the sense of Fredholm in the physically relevant energy spaces. A detailed presentation of these results can be found in the thesis [Sa14]. Publications based on the thesis are in preparation. Curiously, whereas the study of VIE in electromagnetic scattering has thus been completed as far as questions of Fredholm properties are concerned, the simpler case of acoustic scattering does not seem to have been covered in the same depth. It is the purpose of the present paper to close this gap.