Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly

TL;DR

This paper addresses inverse scattering problems for the 3D Hartree equation, demonstrating that under rapid decay conditions, the interaction potential can be uniquely reconstructed and stably estimated from scattering data.

Contribution

It establishes unique determination and stability estimates for the derivatives of the Fourier transform of the potential at zero, based on scattering operator knowledge.

Findings

Unique reconstruction of potential derivatives at zero

Stability estimates for potential identification

Applicable to rapidly decaying potentials

Abstract

We consider inverse scattering problems for the three-dimensional Hartree equation. We prove that if the unknown interaction potential $V(x)$ of the equation satisfies some rapid decay condition, then we can uniquely determine the exact value of $\partial_\xi^\alpha \hat{V}(0)$ for any multi-index $\alpha$ by the knowledge of the scattering operator for the equation. Furthermore, we show some stability estimate for identifying $\partial_\xi^\alpha \hat{V}(0)$.