Darboux integrals for Schrödinger planar vector fields via Darboux transformations
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...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2012 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/16689 |
| Online Access: | https://hdl.handle.net/2117/16689 https://dx.doi.org/10.3842/SIGMA.2012.043 |
| Access Level: | Open access |
| Keyword: | Polynomials Differential equations Darboux theory of integrability Darboux transformations Differential Galois theory Schrödinger equation Supersymmetric quantum mechanics Polinomis Equacions diferencials Classificació AMS::12 Field theory and polynomials::12H Differential and difference algebra Classificació AMS::34 Ordinary differential equations::34A General theory Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::81 Quantum theory::81Q General mathematical topics and methods in quantum theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions diferencials ordinàries |
| Summary: | 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 |
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