Hachem, Ghias The \(\bar \partial\) approach to inverse scattering for Dirac operators. (English) Zbl 0821.35115 Inverse Probl. 11, No. 1, 123-146 (1995). Summary: The inverse-scattering problem for the Dirac operator is studied by the \(\overline {\partial}\) method. We construct the scattering transform associated with a potential \(Q\) and prove that it satisfies a nonlinear \(\overline {\partial}\) equation. We give necessary and sufficient conditions for a matrix-valued function to be the scattering transform of a potential \(Q\). The scattering transform is shown to satisfy a compatibility condition similar to the Newton Miracle condition in the Schrödinger case. Cited in 3 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 35R30 Inverse problems for PDEs 81U40 Inverse scattering problems in quantum theory 35P25 Scattering theory for PDEs Keywords:scattering transform; nonlinear \(\overline {\partial}\) equation; compatibility condition PDFBibTeX XMLCite \textit{G. Hachem}, Inverse Probl. 11, No. 1, 123--146 (1995; Zbl 0821.35115) Full Text: DOI