Summary: Spectral and Condition Number Estimates of the Acoustic
Single-Layer Operator for Low-Frequency Multiple Scattering.
Part I: From Single to Multiple Scattering, Dilute Media
The aim of this paper is to derive asymptotic spectral and condition number estimates of
the acoustic single-layer potential for various low frequency multiple scattering problems. The
obstacles are supposed to be distant (dilute media) to obtain the approximation formulas. We
show that an approach based on the Gershgorin disks provides limited spectral informations. We
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
and for linear boundary element methods with non uniform meshes. This study is completed in
 by new spectral estimates for the case of close obstacles.
1 Introduction 2
2 The Electric Field Integral Equation (EFIE) and the single-layer operator 3
2.1 Single-layer potential and EFIE for multiple scattering . . . . . . . . . . . . . . . . . 3
2.2 Expression of the single-layer potential in the Fourier basis for the multiple scattering