Primitivo B. Acosta-Humánez, Chara Pantazi, “Darboux integrals for Schrödinger planar vector fields via Darboux transformations”, SIGMA, 2012, Volume 8,043, 26 pp.

This article is cited in 7 papers

Darboux integrals for Schrödinger planar vector fields via Darboux transformations

Primitivo B. Acosta-Humánez^a, 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 Politécnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44-50, 08028 Barcelona, Spain

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.

Keywords: Darboux theory of integrability, Darboux transformations, differential Galois theory, Schrödinger equation, supersymmetric quantum mechanics.

MSC: 12H05; 34A30; 34C14; 81Q60; 32S65

Received: March 5, 2012; in final form July 6, 2012; Published online July 14, 2012

Language: English

DOI: 10.3842/SIGMA.2012.043