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 and as a way of constructing soliton solutions of the KdV hierarchy. 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 . The Darboux transformation was rediscovered by physicists as creation and annihilation operators and ladder operators, and is of fundamental importance in supersymmetric quantum mechanics.

History

The idea actually goes back to Carl Gustav Jacob Jacobi.

The 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

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

is a solution of the equation

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

where $Q(x)=q(x)-2\left({\frac {z'(x)}{z(x)}}\right)'.$