Generalizes Darboux transformations: classification of inverse scattering methods for the radial Schrodinger equation

Abstract

A wide class of Darboux transformations of a Sturm-Liouville equation providing a unified treatment of exactly solvable models in quantum scattering theory is considered systematically. A classification of Darboux transformations is given and the relations between Darboux transformations and standard inverse scattering procedures for the radial Schrodinger equation including the matrix method of Newton-Sabatier are studied. In particular, the definitions, properties and matrix generalizations of Darboux transformations associated with Marchenko-type integral equations are studied in detail.