Draft:Darboux transformation

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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 was rediscovered by physicists as creation and annihilation operators and ladder operators, and is of fundamental importance in supersymmetric quantum mechanics^[5].

The idea actually goes back to Carl Gustav Jacob Jacobi.^[6]

Let ${\displaystyle y=y(x)}$ be a solution of the equation

${\displaystyle -y''(x)+q(x)y(x)=\lambda y(x)}$

and ${\displaystyle y=z(x)}$ be a fixed strictly positive solution of the same equation for some ${\displaystyle \lambda =\lambda _{0}}$. Then

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

is a solution of the equation

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

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