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Understanding the spectral properties of boundary integral operators in acoustic scattering has important practical implications, such as for the analysis of the stability of boundary element discretizations or the convergence of iterative solvers as the wave number k grows. Yet, little is known about spectral decompositions of the standard boundary integral operators in acoustic scattering. Theoretical results are mainly available on the unit circle, where these operators diagonalize in a simple Fourier basis. In this paper we investigate spectral decompositions for more general smooth domains. Based on the decomposition of the acoustic Green's function in elliptic coordinates, we give spectral decompositions on ellipses. For general smooth domains we show that approximate spectral decompositions can be given in terms of circle Fourier modes transplanted onto the boundary of the domain. An important underlying question is whether the operators are normal. Based on previous numerical investigations it appears that the standard boundary integral operators are normal only when the domain is a ball and here we prove that this is indeed the case for the acoustic single layer potential. We show that the acoustic single, double and conjugate double layer potential are normal in a scaled inner product on the ellipse. On more general smooth domains the operators can be split into a normal component plus a smooth perturbation. Numerical computations of pseudospectra are presented to demonstrate the nonnormal behaviour on general domains.
Betcke, T., Phillips, J., & Spence, E. A. (2014). Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering. IMA Journal of Numerical Analysis, 34(2), 700-731. https://doi.org/10.1093/imanum/drt002