Essential spectrum of local multi-trace boundary integral operators

TL;DR

This paper derives an exact formula for the spectrum of local multi-trace boundary integral operators in transmission scattering problems, providing insights into their essential spectrum when wave numbers vary, supported by numerical evidence.

Contribution

It introduces a precise analytic formula for the spectrum of these operators in specific geometric and physical settings, extending to variable wave numbers.

Findings

Exact spectrum formula for uniform wave numbers

Identification of the essential spectrum for variable wave numbers

Numerical validation of theoretical spectral results

Abstract

Considering pure transmission scattering problems in piecewise constant media, we derive an exact analytic formula for the spectrum of the corresponding local multi-trace boundary integral operators in the case where the geometrical configuration does not involve any junction point and all wave numbers equal. We deduce from this the essential spectrum in the case where wave numbers vary. Numerical evidences of these theoretical results are also presented.