Singular matrix Darboux transformations in the inverse scattering method

TL;DR

This paper introduces singular Darboux transformations with singular matrix coefficients into the inverse scattering method, providing determinant formulas and a representation for solutions to the Marchenko equation.

Contribution

It extends the inverse scattering method by incorporating singular Darboux transformations and derives explicit determinant formulas for their application.

Findings

Derived determinant formulas for chains of singular Darboux transformations.

Presented a determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation.

Extended the theoretical framework of inverse scattering with singular transformations.

Abstract

Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given.