Determining a Random Schrödinger Operator: Both Potential and Source are Random

Abstract

We study an inverse scattering problem associated with a Schrödinger 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 techniques from the theory of pseudodifferential operators and microlocal analysis.

On recent progress of single-realization recoveries of random Schrödinger systems

“…The results between [21] and [22,25] have two major differences. First, in [21] the random part of the source is assumed to be a Gaussian white noise, while in [22] the potential and the source are assumed to be migr fields. The migr field can fit larger range of randomness by tuning its rough order and rough strength.…”

“…The single-realization recovery has been studied in the literature. In this paper we mainly focus on [8,[18][19][20][21][22][23][24][25].…”

“…Then the authors extended the work [25] to the case where both the potential and the source are random (of migr type), and the extended result is presented in [22]. The results between [21] and [22,25] have two major differences.…”

“…Then the authors extended the work [25] to the case where both the potential and the source are random (of migr type), and the extended result is presented in [22]. The results between [21] and [22,25] have two major differences. First, in [21] the random part of the source is assumed to be a Gaussian white noise, while in [22] the potential and the source are assumed to be migr fields.…”

“…The migr field can fit larger range of randomness by tuning its rough order and rough strength. Second, in [22] both the source and potential are random, while in [25] the potential is assumed to be deterministic. These two facts make [22] much more challenging than that in [25].…”