Free boundary methods and non-scattering phenomena

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.

Reconstruction of inclusions in electrical conductors

We investigate the reciprocity gap functional method, which has been developed in the inverse scattering theory, in the context of electrical impedance tomography. In particular, we aim to reconstruct an inclusion contained in a body, whose conductivity is different from the conductivity of the surrounding material. Numerical examples are given, showing the performance of our algorithm.

Spectral measure function separability and reflectionless potentials

Both reflectionless potentials and special conditions on the spectral measure function have been well studied in inverse scattering theory. This short paper considers a spectral measure function that is separable and shows that it is equivalent to the potential being reflectionless.