×

The \(\bar \partial\) approach to inverse scattering for Dirac operators. (English) Zbl 0821.35115

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.

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
PDFBibTeX XMLCite
Full Text: DOI