Darboux Integrals for Schr\"odinger Planar Vector Fields via Darboux Transformations

TL;DR

This paper explores Darboux transformations of Schrödinger-type planar vector fields, demonstrating how shape invariance preserves integrability structures, with applications to supersymmetric quantum mechanics.

Contribution

It establishes the invariance of Darboux integrability objects under Darboux transformations and highlights the role of shape invariance in preserving vector field structures.

Findings

Darboux transformations preserve integrability objects.

Shape invariance maintains the structure of transformed vector fields.

Applications to supersymmetric quantum mechanics examples.

Abstract

In this paper we study the Darboux transformations of planar vector fields of Schr\"odinger type. Using the isogaloisian property of Darboux transformation we prove the "invariance" of the objects of the "Darboux theory of integrability". In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.