# Determining a random Schr\"odinger operator: both potential and source are random

## TL;DR

This paper addresses an inverse scattering problem for a Schrödinger system with both the potential and source being random, establishing uniqueness results for recovering their rough strengths from single realizations of scattering data.

## Contribution

It introduces novel methods to uniquely recover the rough strengths of both the random source and potential from single realization data using microlocal analysis.

## Key findings

- Single realization of passive scattering recovers source strength.
- Backscattering data determines potential strength.
- Ergodicity enables single realization recovery.

## Abstract

We study an inverse scattering problem associated with a Schr\"odinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered. The ergodicity is used to establish the single realization recovery. The asymptotic arguments in our study are based on the theories of pseudodifferential operators and microlocal analysis.