Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 043, 26 pages      arXiv:1111.0120      http://dx.doi.org/10.3842/SIGMA.2012.043
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

Primitivo B. Acosta-Humánez a and Chara Pantazi b
a) Departamento de Matemáticas y Estadística Universidad del Norte, Km. 5 via Puerto Colombia, Barranquilla, Colombia
b) Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44-50, 08028 Barcelona, Spain

Received March 05, 2012, in final form July 06, 2012; Published online July 14, 2012

Abstract
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.

Key words: Darboux theory of integrability; Darboux transformations; differential Galois theory; Schrödinger equation; supersymmetric quantum mechanics.

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