Stability for the inverse random potential scattering problem

TL;DR

This paper investigates the inverse problem of determining the covariance structure of a Gaussian random potential in Schr"odinger scattering, providing probabilistic stability estimates using ergodic and analytic continuation techniques.

Contribution

It introduces a probabilistic stability estimate for recovering the principal symbol of the covariance operator from single realizations of scattering data.

Findings

Established well-posedness of the direct scattering problem with rough potentials.

Derived a probabilistic stability estimate for the inverse problem.

Used ergodicity and analytic continuation to analyze the stability.

Abstract

This paper is concerned with an inverse random potential problem for the Schr\"odinger equation. The random potential is assumed to be a generalized Gaussian random function, whose covariance operator is a classical pseudo-differential operator. For the direct problem, the meromorphic continuation of the resolvent of the Schr\"odinger operator with rough potentials is investigated, which yields the well-posedness of the direct scattering problem and a Born series expansion. For the inverse problem, we derive a probabilistic stability estimate for determining the principle symbol of the covariance operator of the random potential. The stability result provides an estimate of the probability for an event when the principle symbol can be quantitatively determined by a single realization of the multi-frequency backscattered far-field pattern. The analysis employs the ergodicity theory and quantitative analytic continuation principle.