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Darboux transformation

In mathematics, the Darboux transformation, named after Gaston Darboux (1842–1917), is a method of generating a new equation and its solution from the known ones. It is widely used in inverse scattering theory, in the theory of orthogonal polynomials,[1][2] and as a way of constructing soliton solutions of the KdV hierarchy.[3] From the operator-theoretic point of view, this method corresponds to the factorization of the initial second order differential operator into a product of first order differential expressions and subsequent exchange of these factors, and is thus sometimes called the single commutation method in mathematics literature[4]. The Darboux transformation has applications in supersymmetric quantum mechanics.[5][6]

History

The idea goes back to Carl Gustav Jacob Jacobi.[7]

Method

Let be a solution of the equation

and be a fixed strictly positive solution of the same equation for some . Then for ,

is a solution of the equation

where Also, for , one solution of the latter differential equation is and its general solution can be found by d’Alembert's method:

where and are arbitrary constants.

Eigenvalue problems

Darboux transformation modifies not only the differential equation but also the boundary conditions. This transformation makes it possible to reduce eigenparameter-dependent boundary conditions to boundary conditions independent of the eigenvalue parameter – one of the Dirichlet, Neumann or Robin conditions.[8][9][10][11] On the other hand, it also allows one to convert inverse square singularities to Dirichlet boundary conditions and vice versa.[12][13] Thus Darboux transformations relate eigenparameter-dependent boundary conditions with inverse square singularities.[14]

References

  1. ^ Grünbaum, F. Alberto; Haine, Luc (1996). "Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation". Symmetries and Integrability of Difference Equations. CRM Proc. Lecture Notes. 9. Amer. Math. Soc., Providence, RI: 143–154. doi:10.1090/crmp/009/14. ISBN 978-0-8218-0601-2.
  2. ^ Gómez-Ullate, D; Kamran, N; Milson, R (2010-10-29). "Exceptional orthogonal polynomials and the Darboux transformation". Journal of Physics A: Mathematical and Theoretical. 43 (43): 434016. arXiv:1002.2666. Bibcode:2010JPhA...43Q4016G. doi:10.1088/1751-8113/43/43/434016. ISSN 1751-8113.
  3. ^ Matveev, Vladimir B.; Salle, Mikhail A. (1991-01-01). Darboux Transformations and Solitons. Berlin ; New York: Springer. ISBN 3-540-50660-8.
  4. ^ Deift, P. A. (1978-06-01). "Applications of a commutation formula". Duke Mathematical Journal. 45 (2). doi:10.1215/S0012-7094-78-04516-7. ISSN 0012-7094.
  5. ^ Cooper, Fred; Khare, Avinash; Sukhatme, Uday (2001). Supersymmetry in Quantum Mechanics. World Scientific. Bibcode:2001sqm..book.....C. doi:10.1142/4687. ISBN 978-981-02-4605-1.
  6. ^ Gómez-Ullate, D; Kamran, N; Milson, R (2004-10-29). "Supersymmetry and algebraic Darboux transformations". Journal of Physics A: Mathematical and General. 37 (43): 10065–10078. arXiv:nlin/0402052. doi:10.1088/0305-4470/37/43/004.
  7. ^ Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A. (2010). "Darboux transformations and the factorization of generalized Sturm–Liouville problems". Proceedings of the Royal Society of Edinburgh: Section a Mathematics. 140 (1): 1–29. doi:10.1017/S0308210508000905. ISSN 0308-2105.
  8. ^ Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A. (2002). "Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. I". Proceedings of the Edinburgh Mathematical Society. 45 (3): 631–645. doi:10.1017/S0013091501000773. ISSN 0013-0915.
  9. ^ Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A. (2002). "Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, II". Journal of Computational and Applied Mathematics. 148 (1): 147–168. doi:10.1016/S0377-0427(02)00579-4.
  10. ^ Guliyev, Namig J. (2020). "Essentially isospectral transformations and their applications" (PDF). Annali di Matematica Pura ed Applicata (1923 -). 199 (4): 1621–1648. doi:10.1007/s10231-019-00934-w. ISSN 0373-3114. Archived (PDF) from the original on 2020-05-09. Retrieved 2025-01-12.
  11. ^ Guliyev, Namig J. (2019-06-01). "Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter". Journal of Mathematical Physics. 60 (6). arXiv:1806.10459. doi:10.1063/1.5048692. ISSN 0022-2488. Archived from the original on 2024-09-06. Retrieved 2025-01-12.
  12. ^ Crum, M. M. (1955). "Associated Sturm-Liouville Systems". The Quarterly Journal of Mathematics. 6 (1): 121–127. arXiv:physics/9908019. doi:10.1093/qmath/6.1.121. ISSN 0033-5606.
  13. ^ Carlson, R. (1993). "Inverse Spectral Theory for Some Singular Sturm-Liouville Problems". Journal of Differential Equations. 106 (1): 121–140. doi:10.1006/jdeq.1993.1102.
  14. ^ Guliyev, Namig J (2023-09-14). "Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin". The Quarterly Journal of Mathematics. 74 (3): 889–910. arXiv:2001.00061. doi:10.1093/qmath/haad004. ISSN 0033-5606. Archived from the original on 2024-11-17. Retrieved 2025-01-10.
Quelle: Https://en.wikipedia.org/wiki/Darboux_transformation