Inverse Spectral Analysis of Singular Radial AKNS Operators

TL;DR

This paper investigates the inverse spectral problem for singular radial AKNS operators, establishing local uniqueness for certain parameter pairs and analyzing the properties of the spectral map near zero potential.

Contribution

It provides new results on local uniqueness and injectivity of the spectral map for specific parameter pairs in singular AKNS operators.

Findings

Established local uniqueness for (0,1), (1,2), and (0,3) parameter pairs.

Proved injectivity of the Fréchet differential at zero potential for (0,2).

Open question remains on the closedness of the spectral map's range for (0,2).

Abstract

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter $\kappa\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(\kappa_1,\kappa_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fr\'echet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.