  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.
