A Generalized Method for the Darboux Transformation

TL;DR

This paper introduces a unified generalized method for Darboux transformations that can modify the spectrum of linear differential systems, applicable to various systems and providing explicit formulas for potential and wavefunction changes.

Contribution

It develops a broad, unified approach to Darboux transformations, extending beyond specific systems and deriving explicit formulas for spectral modifications.

Findings

Explicit formulas for potential and wavefunction changes.

Unified approach applicable to various linear systems.

Comparison with standard Darboux methods in special cases.

Abstract

A method is presented to obtain the change in the potential and in the relevant wavefunction of a linear system of ordinary differential equations containing a spectral parameter, when that linear system is perturbed and a finite number of discrete eigenvalues are added to or removed from the spectrum. Some explicit formulas are derived for those changes by introducing certain fundamental linear integral equations for the corresponding unperturbed and perturbed linear systems. This generalized method is applicable in a unified manner on a wide class of linear systems. This is in contrast to the standard method for a Darboux transformation, which is specific to the particular linear system on which it applies. A comparison is provided in some special cases between this generalized method and the standard method for the Darboux transformation. In particular, when a bound state is added to the discrete spectrum, some Darboux transformation formulas are presented for the full-line Schr\"odinger equation, where those formulas resemble the Darboux transformation formulas for the half-line Schr\"odinger equation. The theory presented is illustrated with some explicit examples.