A priori estimates for high frequency scattering by obstacles of arbitrary shape

TL;DR

This paper derives high frequency estimates for scattering operators around arbitrary obstacles and proves that the total scattering cross section is bounded by four times the obstacle's geometrical cross section as frequency increases.

Contribution

It provides the first sharp a priori estimates for high frequency scattering by obstacles of arbitrary shape, linking operator bounds to physical scattering limits.

Findings

Total cross section does not exceed four times the geometrical cross section at high frequencies.

Derived sharp a priori estimates for Dirichlet-to-Neumann and Neumann-to-Dirichlet operators.

Established bounds applicable to obstacles of any shape in high frequency scattering.

Abstract

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number $k\to \infty$. This bound of the total cross section is sharp.