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,{{cite journal | last1 = Grünbaum| first1 = F. Alberto| last2 = Haine| first2 = Luc| title = Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation | journal = Symmetries and Integrability of Difference Equations| series = CRM Proc. Lecture Notes| volume = 9| pages = 143–154| publisher = Amer. Math. Soc., Providence, RI| year = 1996| doi = 10.1090/crmp/009/14| isbn = 978-0-8218-0601-2}}{{cite journal | last1=Gómez-Ullate | first1=D | last2=Kamran | first2=N | last3=Milson | first3=R | title=Exceptional orthogonal polynomials and the Darboux transformation | journal=Journal of Physics A: Mathematical and Theoretical | volume=43 | issue=43 | date=2010-10-29 | issn=1751-8113 | doi=10.1088/1751-8113/43/43/434016 | doi-access=free | page=434016 | arxiv=1002.2666 | bibcode=2010JPhA...43Q4016G }} and as a way of constructing soliton solutions of the KdV hierarchy.{{cite book | last1=Matveev | first1=Vladimir B. | last2=Salle | first2=Mikhail A. | title=Darboux Transformations and Solitons | publisher=Springer | publication-place=Berlin ; New York | date=1991-01-01 | isbn=3-540-50660-8}} 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.{{cite journal | last=Deift | first=P. A. | title=Applications of a commutation formula | journal=Duke Mathematical Journal | volume=45 | issue=2 | date=1978-06-01 | issn=0012-7094 | doi=10.1215/S0012-7094-78-04516-7}} The Darboux transformation has applications in supersymmetric quantum mechanics.{{cite book | last1=Cooper | first1=Fred | last2=Khare | first2=Avinash | last3=Sukhatme | first3=Uday | title=Supersymmetry in Quantum Mechanics | publisher=World Scientific | date=2001 | isbn=978-981-02-4605-1 | doi=10.1142/4687| bibcode=2001sqm..book.....C }}{{Cite journal |last1=Gómez-Ullate |first1=D |last2=Kamran |first2=N |last3=Milson |first3=R |date=2004-10-29 |title=Supersymmetry and algebraic Darboux transformations |url=https://iopscience.iop.org/article/10.1088/0305-4470/37/43/004 |journal=Journal of Physics A: Mathematical and General |volume=37 |issue=43 |pages=10065–10078 |arxiv=nlin/0402052 |doi=10.1088/0305-4470/37/43/004}}

History

The idea goes back to Carl Gustav Jacob Jacobi.{{cite journal | last1=Binding | first1=Paul A. | last2=Browne | first2=Patrick J. | last3=Watson | first3=Bruce A. | title=Darboux transformations and the factorization of generalized Sturm–Liouville problems | journal=Proceedings of the Royal Society of Edinburgh: Section a Mathematics | volume=140 | issue=1 | date=2010 | issn=0308-2105 | doi=10.1017/S0308210508000905 | pages=1–29}}

Method

Let $y = y(x)$ be a solution of the equation

: $-y''(x) + q(x)y(x) = \lambda y(x)$

and $y = z(x)$ be a fixed strictly positive solution of the same equation for some $\lambda = \lambda_0$ . Then for $\lambda \ne \lambda_0$ ,

: $Y(x) := y'(x) - \frac{z'(x)}{z(x)}y(x) = z(x) \left( \frac{y(x)}{z(x)} \right)'$

is a solution of the equation

: $-Y''(x) + Q(x)Y(x) = \lambda Y(x),$

where

$Q(x) = q(x) - 2\left( \frac{z'(x)}{z(x)} \right)'.$ Also, for $\lambda = \lambda_0$ ,

one solution of the latter differential equation is $1/z(x)$ and its general solution can be found by d’Alembert's method:

: $Y(x) = \frac{1}{z(x)} \left( C_1 + C_2 \int^x z^2(x) dx \right),$

where $C_1$ and $C_2$ 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.{{cite journal | last1=Binding | first1=Paul A. | last2=Browne | first2=Patrick J. | last3=Watson | first3=Bruce A. | title=Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. I | journal=Proceedings of the Edinburgh Mathematical Society | volume=45 | issue=3 | date=2002 | issn=0013-0915 | doi=10.1017/S0013091501000773 | doi-access=free | pages=631–645 }}{{cite journal | last1=Binding | first1=Paul A. | last2=Browne | first2=Patrick J. | last3=Watson | first3=Bruce A. | title=Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, II | journal=Journal of Computational and Applied Mathematics | volume=148 | issue=1 | date=2002 | doi=10.1016/S0377-0427(02)00579-4 | doi-access=free | pages=147–168}}{{cite journal | last=Guliyev | first=Namig J. | title=Essentially isospectral transformations and their applications | journal=Annali di Matematica Pura ed Applicata | volume=199 | issue=4 | date=2020 | issn=0373-3114 | doi=10.1007/s10231-019-00934-w | doi-access=free | pages=1621–1648 | url=https://link.springer.com/content/pdf/10.1007/s10231-019-00934-w.pdf | access-date=2025-01-12 | archive-date=2020-05-09 | archive-url=https://web.archive.org/web/20200509125209/https://link.springer.com/content/pdf/10.1007%2Fs10231-019-00934-w.pdf | url-status=live }}{{cite journal | last=Guliyev | first=Namig J. | title=Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter | journal=Journal of Mathematical Physics | volume=60 | issue=6 | date=2019-06-01 | issn=0022-2488 | doi=10.1063/1.5048692 | doi-access=free | arxiv=1806.10459 | url=https://hal.science/hal-01824209/document | access-date=2025-01-12 | archive-date=2024-09-06 | archive-url=https://web.archive.org/web/20240906074702/https://hal.science/hal-01824209/document | url-status=live }} On the other hand, it also allows one to convert inverse square singularities to Dirichlet boundary conditions and vice versa.{{cite journal | last=Crum | first=M. M. | title=Associated Sturm-Liouville Systems | journal=The Quarterly Journal of Mathematics | volume=6 | issue=1 | date=1955 | issn=0033-5606 | doi=10.1093/qmath/6.1.121 | pages=121–127| arxiv=physics/9908019 }}{{cite journal | last=Carlson | first=R. | title=Inverse Spectral Theory for Some Singular Sturm-Liouville Problems | journal=Journal of Differential Equations | volume=106 | issue=1 | date=1993 | doi=10.1006/jdeq.1993.1102 | doi-access=free | pages=121–140}} Thus Darboux transformations relate eigenparameter-dependent boundary conditions with inverse square singularities.{{cite journal | last=Guliyev | first=Namig J | title=Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin | journal=The Quarterly Journal of Mathematics | volume=74 | issue=3 | date=2023-09-14 | issn=0033-5606 | doi=10.1093/qmath/haad004 | doi-access=free | pages=889–910 | url=https://hal.science/hal-02425952/document | access-date=2025-01-10 | archive-date=2024-11-17 | archive-url=https://web.archive.org/web/20241117121259/https://hal.science/hal-02425952/document | url-status=live | arxiv=2001.00061 }}

References

Category:Theoretical physics

Category:Ordinary differential equations