Aktosun, Tuncay; van der Mee, Cornelis A unified approach to Darboux transformations. (English) Zbl 1185.45006 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) Cited in 2 Documents 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 PDF BibTeX XML Cite \textit{T. Aktosun} and \textit{C. van der Mee}, Inverse Probl. 25, No. 10, Article ID 105003, 22 p. (2009; Zbl 1185.45006) Full Text: DOI arXiv Link OpenURL