Using mixed data in the inverse scattering problem

TL;DR

This paper demonstrates that using mixed phase-shift data in inverse scattering problems allows for the unique determination of potentials, leveraging the monotonic properties of zeros of the Schrödinger equation solutions.

Contribution

It introduces a novel approach of combining phase-shift data at fixed angular momentum and energy to achieve unique potential reconstruction in inverse scattering.

Findings

Zeros of regular solutions are monotonic functions of energy.

Mixed data sets enable unique potential determination.

Potential can be reconstructed from combined phase-shift data.

Abstract

Consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$, which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the $r_{n}(E)$ range from zero to infinity. This suggests that the use of the mixed data of phase-shifts $\{\delta(\ell_0,k), k \geq k_0 \} \cup \{\delta(\ell,k_0), \ell \geq \ell_0 \}$, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.