Exceptional circles of radial potentials

TL;DR

This paper investigates the nonlinear scattering transform for 2D Schrödinger equations with radial potentials, revealing explicit examples of singularities linked to non-uniqueness of solutions and exceptional points.

Contribution

It provides the first explicit theoretical and computational examples of singularities in the scattering transform caused by non-uniqueness of complex geometric optics solutions.

Findings

Identified potentials with nontrivial singularities in the scattering transform.

Linked singularities to exceptional points where solutions are non-unique.

Showed the relation between potential types and criticality in the scattering process.

Abstract

A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The singularities arise from non-uniqueness of the complex geometric optics solutions that define the scattering transform. The values of the complex spectral parameter at which the singularities appear are called exceptional points. The singularity formation is closely related to the fact that potentials of conductivity type are critical in the sense of Murata.