Inverse scattering for a random potential

TL;DR

This paper addresses the inverse scattering problem for a random Schrödinger equation, demonstrating that the principal symbol of the covariance operator of a Gaussian random potential can be uniquely identified from backscattered data.

Contribution

It establishes the unique determination of the covariance operator's principal symbol from a single backscattering measurement in a random setting, including the full non-linear case for three dimensions.

Findings

Unique identification of covariance principal symbol from backscattering data

Results applicable to the full non-linear inverse problem in 3D

Practical relevance shown through a physical scaling regime

Abstract

In this paper we consider an inverse problem for the $n$-dimensional random Schr\"{o}dinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential $q$, we uniquely determine the principal symbol of the covariance operator of $q$. Especially, for $n=3$ this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.