Certain Inverse Resonance Uniqueness on the Line with Super-Exponentially Decaying Potential

TL;DR

This paper investigates the inverse resonance problem for rapidly decaying potentials on the line, establishing uniqueness results using complex analysis techniques and resonance data.

Contribution

It introduces a new inverse uniqueness result for fast decaying potentials based on resonance data and Nevanlinna theory.

Findings

Proves inverse uniqueness for potentials from resonance data

Utilizes Froese's Born approximation and Neumann series analysis

Establishes growth estimates of the scattering determinant

Abstract

In the paper, we study the inverse problem with the resonant data of fast decaying potential $V$. We review Froese' construction of the Born's approximation and Neumann series to analyze the growth of scattering determinant. Assuming all the the resonances are given, we deduce the certain inverse uniqueness on $V$ from the Nevanlinna type of representation theorem.