Stability of Inverse Resonance Problem on the Half Line

TL;DR

This paper investigates the inverse resonance problem in one-dimensional scattering, establishing stability estimates for reconstructing potentials from resonance data, especially under regular resonance distributions.

Contribution

It provides quantitative stability estimates for the inverse resonance problem on the half line, linking potential support size to resonance zero distribution and applying value distribution theory.

Findings

Quantitative estimates for zeros and poles of the scattering matrix.

Connection between potential support size and zero distribution.

Stability results for regular resonance distributions.

Abstract

We consider the inverse resonance problem in one-dimensional scattering theory. The scattering matrix consists of $2\times 2$ entries of meromorphic functions, which are quotients of certain Fourier transform. The resonances are expressed as the zeros of Fourier transform of wave field. For compactly-supported perturbation, we are able to quantitatively estimate the zeros and poles of each meromorphic entry. The size of potential support is connected to the zero distribution of scattered wave field. We derive the inverse stability on scattering source based on certain knowledge on the perturbation theory of resonances. When the resonances are distributed regularly, there is certain natural stability through the value distribution theory.