B\"{a}cklund and Darboux transformations for the nonstationary Schr\"{o}dinger equation

TL;DR

This paper investigates how recursive Bäcklund transformations generate new potentials for the nonstationary Schrödinger equation, introduces Darboux transformations for Jost solutions, and analyzes their spectral and analytic properties.

Contribution

It provides a detailed study of potentials generated by Bäcklund transformations and introduces Darboux transformations for Jost solutions, including their spectral data transformations.

Findings

Potentials constructed via recursive Bäcklund transformations are analyzed.

Darboux transformations of Jost solutions are formulated and their properties studied.

Transformations of spectral data are explicitly derived.

Abstract

Potentials of the nonstationary Schr\"{o}dinger operator constructed by means of $n$ recursive B\"{a}cklund transformations are studied in detail. Corresponding Darboux transformations of the Jost solutions are introduced. We show that these solutions obey modified integral equations and present their analyticity properties. Generated transformations of the spectral data are derived.