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Stability for the inverse random potential scattering problem

Tianjiao Wang, Xiang Xu, Yue Zhao

Abstract

This paper is concerned with an inverse random potential problem for the Schrödinger 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 Schrödinger 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.

Stability for the inverse random potential scattering problem

Abstract

This paper is concerned with an inverse random potential problem for the Schrödinger 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 Schrödinger 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.
Paper Structure (8 sections, 12 theorems, 145 equations)

This paper contains 8 sections, 12 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and main result
  3. Direct scattering problem
  4. Stability for the inverse random potential problem
  5. Analysis of the correlation of the zeroth order term in Born series
  6. Analysis of the remaining terms in Born series
  7. Proof of the main theorem
  8. Conclusion

Key Result

Theorem 2.1

Let $V_1,\,V_2 \in \mathcal{V}$ be two random potentials with $h_1,h_2 \in \mathcal{C}_r$ and $\epsilon\in (0, 1/4)$. Furthermore, assume almost surely $\mathbb E\|V_j\|_{W^{\alpha,p}} \le M_3$ with $p \in [ -\frac{3}{\alpha}, +\infty)$ for $\frac{14}{5}<m<3$, and $\mathbb E\|V_j\|_{L^\infty} \le M_ where the positive constant $a(\epsilon) \in (0,1)$ depends on $\epsilon$, and the positive constan

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • ...and 12 more