Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

• Main Authors: Primitivo B. Acosta-Humánez, Chara Pantazi
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2012-07-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: Darboux theory of integrability; Darboux transformations; differential Galois theory; Schrödinger equation; supersymmetric quantum mechanics
• Online Access: http://dx.doi.org/10.3842/SIGMA.2012.043

In this paper we study the Darboux transformations of planar vector fields of Schrödinger 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.