Inverse scattering for a random potential

Abstract

In this paper we consider an inverse problem for the $n$-dimensional random Schrödinger 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.

Previous version 48 pages; Current version 51 pages, 3 figures, several references have been added