Transverse electric scattering on inhomogeneous objects: Singular integral equation, symbol of the operator, and matrix elements

TL;DR

This paper formulates the 2D transverse electric scattering problem as a singular integral equation, derives the operator's symbol for continuous contrasts, and computes matrix elements post-discretization, aiding in understanding and solving electromagnetic scattering.

Contribution

It introduces a rigorous formulation of the TE scattering problem as a singular integral equation and derives the operator's symbol and matrix elements for improved analysis and numerical solutions.

Findings

Derived the symbol of the integral operator for H"older-continuous contrasts.

Calculated matrix elements after discretization using the mid-point rule.

Provided a foundation for analyzing the spectrum and preconditioning of the scattering operator.

Abstract

This is a companion report for the paper "Transverse electric scattering on inhomogeneous objects: Spectrum of integral operator and preconditioning" by the present authors. In this report we formulate the two-dimensional transverse electric scattering problem as a standard singular integral equation, derive the symbol of the integral operator for H\"older-continuous contrasts, and calculate the elements of the system matrix obtained after discretization via the mid-point rule.