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

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.