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On the Riemann–Hilbert problem for the one-dimensional Schrödinger equation. (English) Zbl 0777.34056

Summary: A matrix Riemann-Hilbert problem associated with the one-dimensional Schrödinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann-Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann-Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q40 Bethe-Salpeter and other integral equations arising in quantum theory
34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U40 Inverse scattering problems in quantum theory
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