A unified approach to Darboux transformations. (English) Zbl 1185.45006

Aktosun, Tuncay; van der Mee, Cornelis

Inverse Probl. 25, No. 10, Article ID 105003, 22 p. (2009).

The authors study the one-parameter family of integral equations \[ \beta(x,y) +\zeta(x,y) +\int_x^\infty dz\beta(x,z)w(z,y) =0,\;y>x, \] where \(\beta(x,y) \) is the unknown term and \(w(z,y)\) is a given \(N\times N\) matrix kernel.
Under some conjectures to \(w\), the authors find the resolvent kernel \(r(x,y,z)\). This kernel is written in term of \(\alpha (x,y)\) that is the solution of the equation
\[ \alpha(x,y) +w(x,y) +\int_x^\infty dz \alpha(x,z) w(z,y) =0. \tag{1} \]
The unique solvability to (1) is a priori condition to \(w\).
There are many other results. The proved results allow to sharpen some earlier propositions. The authors give many applications.

Reviewer: Anatoly Filip Grishin (Khar’kov)

MSC:

45F15	Systems of singular linear integral equations
45D05	Volterra integral equations

Keywords:

Marchenko equation; Gel’fand-Levitan equation; Zakharov-Shabat system; involution matrix; degenerate kernel, finite-rank perturbation; wave function; Darboux transformations; resolvent kernel