Spectral and condition number estimates of the acoustic single-layer operator for low-frequency multiple scattering in dilute media

The aim of this paper is to develop an analysis of the distribution of the eigenvalues of the acoustic single-layer potential for various low frequency two-dimensional multiple scattering problems. The obstacles are supposed to be distant (dilute media). In [25], it is shown that an approach based on the Gershgorin disks provides limited spectral information. We therefore introduce an alternative approach by applying the power iteration method to the limit matrix (associated with the zero order spatial modes) which results in satisfactory estimates. All these approximations are built for circular cylinders and formally extended to ellipses and rectangles for linear boundary element methods with non uniform meshes. This study is completed in [26] by spectral estimates for the case of close obstacles.