A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy

TL;DR

This paper develops a Riemann-Hilbert problem approach to inverse scattering for the 2D Schrödinger equation at fixed energy, allowing for potentials with singularities and applying to the Novikov-Veselov equation.

Contribution

It introduces a global Riemann-Hilbert framework for inverse scattering at fixed energy, accommodating non-small potentials and Faddeev exceptional points.

Findings

Established a global inverse scattering method for 2D Schrödinger equation.

Allowed for potentials with singularities and exceptional points.

Applied results to the Novikov-Veselov equation.

Abstract

We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or generic versions of the key results of this approach for the case of positive energy and compactly supported potentials. In particular, we do not assume that the potential is small or that Faddeev scattering solutions do not have singularities (i.e. we allow the Faddeev exceptional points to exist). Applications of these results to the Novikov-Veselov equation are also considered.