Spectral properties of the Dirichlet-to-Neumann operator for exterior Helmholtz problem and its applications to scattering theory

TL;DR

This paper proves the spectral properties of the Dirichlet-to-Neumann operator for exterior Helmholtz problems and applies these findings to derive bounds in scattering theory, improving understanding and numerical approximation accuracy.

Contribution

It establishes that the Dirichlet-to-Neumann operator has no spectrum in the lower half-plane and applies this to scattering problems with impedance boundary conditions.

Findings

No spectrum of DtN in the lower half-plane

Upper bounds for scattering amplitude gradient and total cross section

Bounds on numerical approximation errors without unknown constants

Abstract

We prove that the Dirichlet-to-Neumann operator (DtN) has no spectrum in the lower half of the complex plane. We find several application of this fact in scattering by obstacles with impedance boundary conditions. In particular, we find an upper bound for the gradient of the scattering amplitude and for the total cross section. We justify numerical approximations by providing bounds on difference between theoretical and approximated solutions without using any a priory unknown constants.