Free boundary methods and non-scattering phenomena

TL;DR

This paper investigates conditions under which penetrable obstacles in inverse scattering can have incident waves that do not scatter, revealing geometric and boundary regularity properties linked to free boundary problems.

Contribution

It establishes the existence of non-scattering incident waves for real-analytic boundary obstacles and characterizes boundary regularity and thinness conditions using free boundary theory.

Findings

Existence of non-scattering waves for real-analytic obstacles.

Obstacles with inward cusps can admit non-scattering waves at zero frequency.

Boundary regularity or thinness is necessary for non-scattering properties.

Abstract

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.